Computational Interface Capturing Methods for …Computational Interface Capturing Methods for Phase Change in Porous Media by Lloyd James Bridge M.Math. (Mathematics) University of

Computational Interface Capturing Methods

for Phase Change in Porous Media

by

Lloyd James Bridge

M.Math. (Mathematics) University of East Anglia, 1998M.Sc. (Applied Mathematics) University of British Columbia, 2002

A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

in

The Faculty of Graduate Studies

(Mathematics)

The University Of British Columbia

November 7, 2006

c© Lloyd James Bridge 2006

ii

Abstract

In this thesis, computational interface capturing methods for mathematicalmodels related to fluid phase change processes in porous media are stud-ied. The mathematical models are often singular and degenerate, whichcontributes to the computational difficulty.

An analysis of a smoothing method applied to a one dimensional freeinterface problem is presented. An asymptotic analysis shows the dependenceof the error in the computed interface location on the chosen small smoothingradius.

Numerical convergence studies are performed for existing capturing meth-ods applied to simple, scalar, moving interface problems, for later comparisonwith convergence rates for a new capturing method applied to a coupled, vec-tor model problem.

A model problem for two-phase fluid flow and heat transfer with phasechange in a porous medium is described. The model is based on a steam-water mixture in sand. Under certain conditions, a two-phase zone, in whichliquid and vapour coexist, is separated from a region of only vapour by aninterface. Two numerical methods are described for locating the interfacein the one-dimensional, steady-state problem; one of these is based on anexisting method, while the other uses the method of Residual Velocities.Agreement between solutions from these two methods is demonstrated, andthe results from the steady-state computations are used as benchmarks forthe numerical results for the transient problem.

It is shown that methods such as front-tracking and the level-set methodare not practical for the solution of the transient problem, due to the indeter-minate nature of the interface velocity, in common with similar degeneratediffusion problems. An interface-capturing method, based on a two-phasemixture formulation, is presented. A finite volume method is developed, andnumerical results show evolution to the correct steady-state. Furthermore,similarity solutions are found, and the interface is shown to propagate at thecorrect velocity, by way of a numerical convergence study. Numerical results

Abstract iii

for the two-dimensional problem are also presented.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction and Background . . . . . . . . . . . . . . . . . . . 11.1 Free and moving interface problems . . . . . . . . . . . . . . . 21.2 Numerical methods for interface problems . . . . . . . . . . . 61.3 Two-phase flow with phase change in porous media . . . . . . 81.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Solutions to prototype interface problems . . . . . . . . . . . 162.1 The steady-state, two-phase Stefan problem . . . . . . . . . . 16

2.1.1 The one-dimensional problem . . . . . . . . . . . . . . 162.1.2 The two-dimensional problem . . . . . . . . . . . . . . 182.1.3 A smoothing method and asymptotic results . . . . . . 222.1.4 The method of residual velocities . . . . . . . . . . . . 27

2.2 The time-dependent, two-phase Stefan problem . . . . . . . . 292.2.1 Mathematical formulation . . . . . . . . . . . . . . . . 302.2.2 Analytical solutions . . . . . . . . . . . . . . . . . . . . 322.2.3 Formulation of the enthalpy method . . . . . . . . . . 362.2.4 Mathematical justification for the enthalpy method . . 422.2.5 Numerical results and convergence study . . . . . . . . 45

2.3 The Porous Medium Equation . . . . . . . . . . . . . . . . . . 492.3.1 Examples and applications . . . . . . . . . . . . . . . . 492.3.2 Analytical solutions . . . . . . . . . . . . . . . . . . . . 51

Contents v

2.3.3 Numerical methods . . . . . . . . . . . . . . . . . . . . 552.3.4 Numerical results and convergence study . . . . . . . . 58

3 Nonisothermal, steady-state phase change in porous media 603.1 Mathematical formulation of the model problem . . . . . . . . 613.2 An iterative disjoint-domain method . . . . . . . . . . . . . . 68

3.2.1 Numerical results and discussion . . . . . . . . . . . . . 713.3 The method of residual velocities . . . . . . . . . . . . . . . . 75

3.3.1 Numerical results and discussion . . . . . . . . . . . . . 763.3.2 A model problem in higher dimensions . . . . . . . . . 76

3.4 Benchmark solutions . . . . . . . . . . . . . . . . . . . . . . . 79

4 The M2 mixture method for the transient Udell problem . 804.1 Mathematical formulation of the model problem . . . . . . . . 80

4.1.1 Modified Stefan conditions at the interface z = L(t) . . 824.1.2 Model summary . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Fixed domain, mixture formulation . . . . . . . . . . . . . . . 874.3 Computational method . . . . . . . . . . . . . . . . . . . . . . 894.4 Numerical results and discussion . . . . . . . . . . . . . . . . . 934.5 Similarity solutions and numerical convergence study . . . . . 95

4.5.1 A reduced model problem . . . . . . . . . . . . . . . . 964.5.2 Travelling wave solution for case psat(T ) = αT . . . . . 994.5.3 Numerical results and convergence study . . . . . . . . 1044.5.4 Other choices for psat(T ) . . . . . . . . . . . . . . . . . 107

4.6 Computations in higher dimensions . . . . . . . . . . . . . . . 1074.6.1 Mathematical model . . . . . . . . . . . . . . . . . . . 1084.6.2 Numerical results and discussion . . . . . . . . . . . . . 109

5 Conclusions and future work . . . . . . . . . . . . . . . . . . . 1175.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.1 The “Big-H” regularisation . . . . . . . . . . . . . . . . 119

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

vi

List of Tables

2.1 Constants for the Stefan problem . . . . . . . . . . . . . . . . 452.2 Errors for the enthalpy method, Forward Euler time-stepping,

with c = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.3 Errors for the enthalpy method, Forward Euler time-stepping,

with c = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4 Errors for the enthalpy method, Forward Euler time-stepping,

for a Neumann problem. . . . . . . . . . . . . . . . . . . . . . 492.5 Errors for Evje’s method, Forward Euler time-stepping, for the

Porous Medium Equation on −0.5 < x < 0.5, with µ = 0.1and c = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.6 Errors for Evje’s method, Forward Euler time-stepping, for thePorous Medium Equation on −0.5 < x < 0.5, with µ = 0.5and c = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1 Constants for Udell problem . . . . . . . . . . . . . . . . . . . 633.2 The effect of varying boundary temperature. . . . . . . . . . . 733.3 The effect of increasing mass. . . . . . . . . . . . . . . . . . . 74

4.1 Errors for reduced Udell problem, implicit (BE) time-steppingwith µ = 0.2, c = 4. . . . . . . . . . . . . . . . . . . . . . . . . 105



vii

List of Figures

1.1 The temperature profile for the solution of the steady-stateStefan problem (1.3)-(1.4). Here, we have taken T0 = −1, T1 =1, Kice = 2.2, and Kwater = 0.55. . . . . . . . . . . . . . . . . 4

1.2 Geothermal power generation. . . . . . . . . . . . . . . . . . . 101.3 A PEM fuel cell. . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 A three-zone system for Udell’s experiment. . . . . . . . . . . 13

2.1 The steady-state, two-phase Stefan problem in two dimensions. 192.2 Temperature profile and contours for steady-state two-phase

Stefan problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Temperature profile and contours for a steady-state two-phase

Stefan problem with multiple interfaces. . . . . . . . . . . . . 212.4 Smooth approximations converging to exact, nonsmooth steady-

state temperature (dots) as ε→ 0, for Example 2.1. . . . . . . 242.5 Computed smooth approximation and asymptotic form, for

Example 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 A sketch of a monotonic increasing smoothed Heaviside func-

tion, Hε. Clearly,∫ 0

XHε(ξ) dξ = O(ε), for X < 0. . . . . . . . 26

2.7 The steady-state, one-dimensional Stefan problem (2.1)-(2.2). 272.8 Convergence to steady-state interface location, using Residual

Velocity computations. . . . . . . . . . . . . . . . . . . . . . . 292.9 Heat balance at the freezing interface - the Stefan condition. . 312.10 A typical Neumann solution to the freezing problem (2.32)-

(2.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.11 A typical travelling wave solution to the freezing problem (2.37). 352.12 Enthalpy as a function of temperature. . . . . . . . . . . . . . 372.13 Enthalpy and smoothed enthalpy as functions of temperature. 412.14 The (z, t)-plane for the enthalpy formulation. . . . . . . . . . . 422.15 Evolution of numerical results using the enthalpy method. . . 46

List of Figures viii

2.16 Evolution of the interface using the enthalpy method, togetherwith associated Neumann result. . . . . . . . . . . . . . . . . . 47

2.17 Temperature history at the point z = 0.5. . . . . . . . . . . . 482.18 Computed interface location for travelling wave conditions,

with c = 1 and c = 3. . . . . . . . . . . . . . . . . . . . . . . . 492.19 A thin droplet spreading under gravity over a solid substrate. 512.20 A solution of the steady-state Porous Medium Equation. . . . 522.21 A solution of the time-dependent Porous Medium Equation. . 532.22 Barenblatt-Pattle solutions of the Porous Medium Equation. . 552.23 Evolution of profiles given by conservative and non-conservative

schemes applied to ut = (u3ux)x. . . . . . . . . . . . . . . . . . 58

3.1 Udell’s experiment. . . . . . . . . . . . . . . . . . . . . . . . . 603.2 A two-zone system for Udell’s experiment. . . . . . . . . . . . 623.3 System for steady-state solution of the free interface problem. 693.4 Solution profiles for free boundary problem using the iterative

disjoint-domain method. . . . . . . . . . . . . . . . . . . . . . 723.5 Close-up of temperature profile for free boundary problem us-

ing the iterative disjoint-domain method. . . . . . . . . . . . . 733.6 Condensation rate in the two-phase zone. . . . . . . . . . . . . 743.7 Convergence to correct interface location, using method of

residual velocities, for T0 = 320, T1 = 450, W = 30. . . . . . . 763.8 Convergence to correct interface location, using method of

residual velocities, for T0 = 375, T1 = 500, W = 36. . . . . . . 773.9 Steady-state system in higher dimensions. . . . . . . . . . . . 78

4.1 Mass conservation at the moving interface. . . . . . . . . . . . 834.2 Temperature profile and the moving interface. . . . . . . . . . 854.3 Grid and staggered grid. . . . . . . . . . . . . . . . . . . . . . 914.4 Evolution to correct steady-state. . . . . . . . . . . . . . . . . 944.5 Interface location L(t), with grid refinement. . . . . . . . . . . 954.6 Temperature history at a point. . . . . . . . . . . . . . . . . . 964.7 Timestep and interface location. . . . . . . . . . . . . . . . . . 974.8 Travelling wave profiles for mixture density, temperature and

saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.9 Computed and exact interface location for the reduced travel-

ling wave problem. . . . . . . . . . . . . . . . . . . . . . . . . 1044.10 Steady-state, one-dimensional profiles. . . . . . . . . . . . . . 109

List of Figures ix

4.11 Saturation and temperature evolution. Saturation contoursare in the top row, temperature contours are in the bottom row.110

4.12 Saturation and temperature profiles - initial condition andlong time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.13 Saturation contours with liquid and vapour fluxes. . . . . . . . 1124.14 Saturation contours for an initial “blob” of two-phase fluid in

the centre of the domain. . . . . . . . . . . . . . . . . . . . . . 1134.15 Long time saturation and temperature profiles at fixed x for

the spreading, migrating blob, together with one-dimensional,steady-state solution. . . . . . . . . . . . . . . . . . . . . . . . 114

4.16 Saturation contours for an initial condition with multiple two-phase regions in the centre of the domain. . . . . . . . . . . . 115

4.17 Long time saturation and temperature profiles at fixed x forinitial multiple two-phase zone problem, together with one-dimensional, steady-state solution. . . . . . . . . . . . . . . . . 116

x

Acknowledgements

This thesis was completed under the supervision of Brian Wetton. I wishto thank Brian for his support, guidance, kindness, and most of all, his en-couragement. I am also grateful for the helpful comments of my supervisorycommittee, Roger Beckie, Chen Greif, Brian Seymour and Michael Ward.

I have been lucky to work with members of the MITACS MathematicalModeling and Scientific Computation group. In particular, I would like tothank Radu Bradean, Atife Caglar, Clarina Chan, Wan Chen, Roger Donald-son, Arian Novruzi, Keith Promislow, Akeel Shah, John Stockie, JF Williamsand Jianying Zhang, for all their help and support.

I am very grateful for the guidance and supervision given by HuaxiongHuang, during two summers spent at York University. The code for the two-dimensional computations presented in this thesis was written during a veryproductive two weeks spent at the University of Saskatchewan. I wish tothank Ray Spiteri and his group there for their work and for making me feelso welcome.

The support of friends and family has been so important, and I am forevergrateful to them all. Special thanks go to Jason Mitchell and Johnson Go fortheir friendship, support, and rescuing skills during my time in Vancouver.

This work was in part funded by MITACS and Ballard Power Systems.My time in Saskatchewan was funded through the MITACS Mobility Fund.

1

Chapter 1

Introduction and Background

Accurate numerical prediction tools and simulations are required for manyphysical processes which arise in industrial and environmental applications.The starting point for developing such numerical methods is the mathemat-ical statement of the physical laws which must be obeyed. This typicallyinvolves one or more partial differential equations, which must be satisfiedon the domain of interest. The mathematical model is completed by thespecification of conditions on the boundaries of the domain and on any inter-faces between regions inside the domain, and the initial state of the system.The solution of the mathematical model for its dependent variables formsthe basis for a numerical simulation.

It is very rare that a mathematical model for a complex physical processyields a closed form solution. As such, a numerical approximation to theexact solution is usually sought. The development of an algorithm to find agood approximation relies in part on the careful implementation of the cor-rect boundary and interface conditions. This can be a nontrivial task for aproblem which is posed on a domain with fixed, known boundaries. Clearly,both the correct statement of a mathematical model, and the subsequent nu-merical procedure, become more complicated if there are moving boundariesor interfaces. The location of any free or moving boundary becomes an extraunknown in the model, and an extra condition is required.

Free and moving boundary problems have attracted an enormous amountof interest in recent years. In this chapter, we give a brief introduction to freeand moving boundary problems, and methods available for their numericalsolution. We then discuss applications and model problems arising in thestudy of phase change, and in particular, two-phase flow with phase changein porous media, which is the focus of the work in this thesis.

In this thesis, we formulate model problems for phase change in porousmedia, and develop numerical methods for the solution of such problems.These models and methods represent a contribution to the literature, in thatthey avoid many of the simplifying assumptions and computational regulari-

Chapter 1. Introduction and Background 2

sations that we will describe along the way, and we have demonstrated theirvalidity by way of convergence studies using analytical solutions which wehave constructed.

1.1 Free and moving interface problems

Many problems from applied science involve interfaces which separate do-mains in which different physical processes or flow regimes occur. The prob-lem of a liquid droplet spreading under gravity concerns the interface betweenthe liquid and the surrounding air, and the moving contact line which sep-arates the wetted area beneath the droplet and the dry substrate ahead ofit. Nonlinear hyperbolic equations represent gas dynamic quantities suchas the gas velocity in a tube, where a region of moving gas and a regionof stationary gas are separated by a moving interface, or shock. A meltingblock of ice contains both a region of liquid water, and a region of solid ice,and we conceive of a sharp interface between the two regions. We will referto the problem of locating such an interface, which is stationary, as a freeboundary problem or free interface problem. Correspondingly, we referto the problem of finding a moving interface as a moving boundary prob-lem, or moving interface problem. Many heat transfer and fluid dynamicprocesses result in free and moving boundary problems, and as such, theseproblems have generated much interest through industrial applications. An-alytical progress in the study of these complex industrial problems is oftenlimited to asymptotic results and similarity solutions for idealised physicalsituations. As such, the primary tool for solution of these nonlinear problemsis usually numerical computation. However, the analysis of certain free andmoving boundary problems has received attention, and is well understood.The prototype Stefan problem for phase change is well documented in theliterature, in particular in [3, 20]. Further, the development of existing nu-merical methods for free and moving boundary problems has often relied onthe mathematical insight into such problems.

Classical solutions to free boundary problems are not usually available.In models where physical parameters are discontinuous across an interface,solutions will lack regularity at the interface. Consider, for example, theone-dimensional, steady-state, free interface problem for a partially meltedblock of ice. Suppose the ice-water system occupies a domain z ∈ [0, D], andthat an interface z = s lies between a region of ice (0 < z < s) and a region


of liquid water (s < z < D). Assuming that the liquid is stationary, thetemperature T throughout the system is given by the mathematical model:

(KiceTz)z = 0 0 < z < s,T (0) = T0 (< 0),T (s−) = 0,

(KwaterTz)z = 0 s < z < D,T (s+) = 0,T (D) = T1 (> 0),

(1.1)

together with the heat balance at the interface

−[KTz

]s+s−

= −(KwaterTz

∣∣∣z→s+

− KiceTz

∣∣∣z→s−

)= 0. (1.2)

The problem of locating the phase-change interface, and determining thetemperature throughout the domain, is commonly referred to as a Stefanproblem, and so we shall call the system (1.3)-(1.4) the mathematical for-mulation of a Steady-State Stefan Problem. Here, the known meltingtemperature is zero, and K is a thermal conductivity. The steady-state po-sition, s, of the melting interface is given by

s =

(T0Kice

T0Kice − T1Kwater

)D. (1.3)

Then the temperature profile is given by

T (z) =

−T0

sz + T0 0 ≤ z ≤ s,

T1

D−s(z −D) + T1 s ≤ z ≤ D.(1.4)

In Figure 1.1, we show a temperature profile for the solution with T0 =−1, T1 = 1, Kice = 2.2, Kwater = 0.55, and D = 1. While the temperatureis continuous throughout the entire domain 0 < z < D, it is clear that itsderivative is not continuous across the interface z = s. For the correspond-ing time-dependent Stefan problem, the temperature profile may evolve witha discontinuous derivative at the interface. As such, if we are able to re-formulate such a problem as a PDE for temperature over the fixed domain0 < z < D, then classical solutions will, in general, be unavailable. Thatis, we will not be able to find solutions, with continuous derivatives up to


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z

Tem

pera

ture

, T

Steady−state temperature profile for 1D, two−phase Stefan problem

ICE

LIQUIDWATER

interface s=0.8

Figure 1.1: The temperature profile for the solution of the steady-state Ste-fan problem (1.3)-(1.4). Here, we have taken T0 = −1, T1 = 1, Kice =2.2, and Kwater = 0.55.

the order of the PDE, which satisfy the PDE at all points in the domain.If the conditions under which a “solution” is defined are weakened, then wemay admit solutions with discontinuous derivatives, as so-called weak, orgeneralised solutions. Such solutions need only satisfy an integral form ofthe original PDE. We will discuss this concept in detail in Chapter 2, in ourtreatment of the so-called enthalpy method for numerical solution of phase-change problems. Weak formulations for free and moving boundary problemsare presented in detail by Elliott and Ockendon [23], as well as others [3, 20].In fact, it is the weak formulation of the Stefan problem which provides abasis for the mathematical justification of the enthalpy method.

Another moving interface problem of physical interest may be describedmathematically by the Porous Medium Equation. This is a nonlineardiffusion equation whose diffusion coefficient is a power of the dependentvariable. Mathematically, we have

ut = ∇. (un∇u), (1.5)

where n > 0, and t is time. Isothermal, ideal gas transport through porousmedia results in (1.5) with n = 1, where u represents the gas density (see,for example, [74]). The spreading of a thin liquid droplet on a solid substrateunder the effect of gravity is described by (1.5) with n = 3 [23]. In this case,


u is the height of the free surface of the droplet.The behaviour of solutions to equation (1.5) is significantly different to

that of linear diffusion equations. A solution of the equation (1.5) is said tohave compact support if u(x, t) vanishes outside a region of x, which varieswith t. The region for which the solution is non-zero is called the support of u.In contrast with solutions of linear diffusion equations, equation (1.5) has theproperty that, for an initial condition u(x, 0) with compact support, the so-lution u(x, t) also has compact support, for t > 0. The “interface”, whichis the boundary of the support of u(x, t), will move with finite speed. Also,this speed may be zero until the solution at the interface has adjusted to aparticular structure required for motion, giving so-called “waiting-time” solu-tions [32, 43]. Again, any interface-type solutions that we construct for (1.5)are understood to be weak, or generalised, solutions. The equation will not,in general, be satisfied by a constructed solution at the interface. Further-more, singular gradients may appear at the interface, where the diffusioncoefficient vanishes, and the equation degenerates. That is, equation (1.5)ceases to be parabolic at the boundary of its support. We will discuss someimplications of the singularity and degeneracy of such problems further inChapter 2.

There is an enormous number of other free and moving boundary prob-lems which we will not describe here. The interface between liquid and thesurrounding air in Hele-Shaw flow [53], and the free liquid-air surface createdby seepage of liquid through a porous dam are two well-studied examples,which bear some relation to our work. Crank’s book [20] presents a varietyof other free and moving boundary problems. However, our discussion willlargely concentrate on the two prototype examples of Stefan problems andthe Porous Medium Equation, which provide a good starting point for themathematical detail of the interface problems discussed in this thesis. Theseproblems may be extended, generalised and combined to model the morecomplex physical problems in which we are particularly interested here.

While the few analytical solutions which are available for free and movingboundary problems may only be applicable to a narrow range of physicalproblems, the insight gained from them is often valuable. Furthermore, theperformance of the numerical methods which are developed for the solutionof such problems is typically evaluated by way of convergence study usingtest cases with analytical solutions.


1.2 Numerical methods for interface

problems

The formulation of an initial-boundary value problem requires the specifica-tion of boundary conditions. In a free or moving interface problem, one ormore conditions are therefore required on the interface, whose position is anunknown in the problem. In particular, in a moving interface problem, atleast one of these conditions will be on the interface velocity, and a success-ful numerical scheme must accurately capture this velocity. Front-trackingmethods (see [20], Ch. 4) are those which explicitly compute the interfacevelocity at each time step, and use this velocity to advance the interface.The use of fixed, uniform grids is not possible for such methods, since specialdifference formulae are required in the vicinity of the interface, as the inter-face will not, in general, coincide conveniently with a grid point. Crank [20]describes various approaches to solving this problem, including the modifieddifference formulae, and adaptive time-stepping to ensure the coincidence ofthe interface with grid nodes. He also presents the related approach of front-fixing. Here, a coordinate transformation is performed after each update ofthe interface location, to ensure that the interface falls at a fixed coordinatevalue. Therefore, this method involves not only the implementation of theinterface velocity at each time step, but also remeshing at each time step.

Certain computational challenges become apparent when considering front-tracking and front-fixing methods. Both of these approaches require a numer-ical implementation of the interface velocity. In degenerate diffusion prob-lems such as the porous medium equation, the interface velocity is often seento be the limit of an indeterminate form (see [43], and [54] Ch 7 ). Numericalmethods are available for such scalar problems [25], which avoid the need forexplicit implementation of the interface velocity, but in more complex prob-lems to which these methods do not extend, front-tracking type methodswould be impractical unless the indeterminate velocity could be accuratelycomputed. Also, in greater than one dimension, the solution of boundaryvalue problems on either side of the interface requires consideration of theinterface geometry, and hence, unstructured grids and coordinate transfor-mations are common. Donaldson [22] presents a front-tracking approach toa generalised Stefan problem using the finite element method in two dimen-sions. Representation of the interface as a spline contributes to a lengthycomputation. For many Stefan-type problems, front-tracking methods have


been avoided.There has been much activity in recent years in developing methods for

moving interface problems which avoid the need for explicit tracking of theinterface. A method which computes the solution to a free or moving inter-face problem without explicitly computing the interface location, but ratherrecovers the interface location from the numerical solution to a reformulatedproblem, is generally referred to as a front-capturing method. The en-thalpy method for the Stefan problem is described in detail by a numberof authors [3, 20, 23, 54], due to its wide applicability. The method takesa reformulation of the problem based on a transformation of the dependentvariable from temperature to enthalpy, leaving a problem for the enthalpyand Kirchoff temperature over a fixed domain which contains the phase-change interface. The Stefan condition for the interface velocity is implicitlyabsorbed into the fixed domain formulation. Finite difference and finite vol-ume schemes for the enthalpy method typically result in a lower order of ac-curacy than corresponding front-tracking schemes [24, 31], but the enthalpymethod is widely used in practice, due to its ease of implementation (see, forexample, [11, 24]). Since the method does not require any explicit handlingof the interface, its extension from one to two or more dimensions is straight-forward. In Chapter 2, we will describe the enthalpy method for the Stefanproblem in detail.

An alternative method which has been developed more recently is theLevel Set Method [55, 61]. This is also a capturing method which results froma reformulation over a fixed domain containing an interface. Rather thanthe interface velocity being implicitly absorbed into the new formulation, itappears explicitly in a Hamilton-Jacobi type equation. While the Level SetMethod does not require a calculation of the interface location at each timestep, as in front-tracking methods, it does require an accurate implementationof the interface speed function. In [18], Chen et al present a Level Set methodfor solving a Stefan problem. Their computation is more expensive thanan enthalpy method for the same problem, but higher order of accuracy isachieved.

The Immersed Interface Method is a modern numerical method whichmay be used for certain moving interface problems. Like the enthalpy method,it is a fixed grid method, but one which explicitly incorporates the interfaceconditions into the discrete scheme. Correction terms are added to the dis-cretization in the neighbourhood of the interface, and interface conditionsmay be implemented by way of a smoothed delta function centered on the


interface. Li [46] describes the method in some detail, and presents nu-merical convergence studies for several example problems. In particular, hedemonstrates second order accuracy when the method is applied to a two-phase Stefan problem. Careful choices for the correction terms and smootheddelta function can result in second order accuracy, in cases where the cor-responding implementation of the enthalpy method can only achieve firstorder accuracy. Hou et al [36] present a hybrid method which combines theImmersed Interface and Level Set methods for interface problems, payingparticular attention to achieving second order accuracy. However, for rela-tive ease of implementation, the enthalpy method often appears to be thepreferred method of solution for simple Stefan problems.

Conservation of mass is an important concept in the development of nu-merical schemes for moving interface problems. It is well known that finitedifference schemes for hyperbolic conservation laws with shock-type solutionsmust be derived in conservative form in order for the shock to propagate atthe correct speed (see [44], for example). It is not enough for a scheme simplyto be consistent with the governing PDE, when non-unique weak solutionsexist. Critically, the conserved “mass” in the continuous physical problemmust also be conserved in time in the discrete approximation. The necessityfor discrete conservation is also a feature of numerical methods for equationsof degenerate parabolic type [25, 26], which model moving interfaces whichare not necessarily shocks.

1.3 Two-phase flow with phase change in

porous media

An understanding of heat transfer and fluid flow through porous media iscentral to the analysis of many environmental and technological processes.Soil is one of many geological materials that is both porous and permeablefor the liquids and gases with which it naturally interacts. Waste disposal of-ten requires fluid transport through a porous structure, as does oil recovery,where thermal effects may also be significant. The design of porous insu-lation clearly raises issues of heat transfer and the migration of moisture.The modelling of these and other such processes must take into account thegeometry of the porous solid, which impedes the flow of the fluid throughthe medium.


The problem of modelling heat and mass transfer of a single-phase fluidflowing through a porous medium is somewhat challenging. A significantlymore complicated modelling problem concerns the heat and mass transfer,and phase change, of a fluid which flows through a porous structure. Two-phase flow and transport with phase change in porous media has attractedmuch interest in recent years, from researchers in such diverse fields as mi-crowave heating of foods and biomaterials [52], geothermal energy recov-ery [59, 74], and fuel cell technology [58, 21]. The process of wood dryingis examined in detail by Whitaker [72]. In these examples, it is importantto consider not only the flow and heat transfer, but also the effect of phasechange, which further complicates the modelling and numerical effort. In con-figurations involving condensation and evaporation, regions of single-phasefluid and two-phase fluid often coexist within the porous medium. The lo-cation of interfaces between single-phase and two-phase zones are often ofprimary interest.

Woods [73] analyzes the processes involved in liquid injection into a hotporous medium, as related to models of geothermal reservoirs. Typically,cool liquid water is pumped into a porous superheated reservoir below theEarth’s surface. The injected water boils at a front, and the resulting watervapour is extracted by a well in order to drive turbines for the generation ofelectricity (see Figure 1.2). In [73], the flow of vapour ahead of the boilingfront is analyzed, and also the flow of liquid behind the front. The rate ofvapour recovery through the well depends on the rate of migration of theboiling front, which in turn depends on the liquid temperature and injectionrate.

While two-phase flow and phase change in geothermal reservoirs arisethrough the injection of cold liquid into a hot porous solid, similar flow phe-nomena arise in certain oil recovery processes. Typically, a displacing fluidis pumped into an underground reservoir, in order to extract oil from theporous rock. Allen et al [5] look in detail at the mathematical modelling ofsuch flows, with particular emphasis on reservoir simulation. Peaceman [57]presents a comprehensive introduction to modelling and numerical methodsfor reservoir simulation, and Allen [4] reviews numerical methods for isother-mal flows in natural porous media. A number of reductions to the mathemat-ical model of multiphase flow in porous media are made under assumptionsspecific to reservoir flow. Flows are typically taken as isothermal, with nophase change throughout the reservoir, and capillary effects are assumed tobe negligible. Under such assumptions, the fluids are incompressible, and


hot porous reservoir

cool liquid

pumped down

hot vapour extracted

through well

Pump Turbine power station

Figure 1.2: Geothermal power generation.

the system of PDEs governing the flow decouples, and reduces to an ellipticequation for the total pressure, and a hyperbolic equation for the saturationof the wetting phase. These simplified problems may be solved numericallyby a well established sequential Implicit Pressure, Explicit Saturation (IM-PES) time-stepping method (see [57]). Karlsen et al [38] take an alternativeapproach, by using a fast marching Level Set method for the saturation equa-tion. The same assumptions of isothermal, incompressible flow without phasechange are key to their formulation.

Oil recovery processes can be thermally enhanced, whereby the injectedfluid is a hot vapour such as steam. This acts to transfer heat to the oil, low-ering its viscosity, which increases its mobility. Clearly, thermal and phasechange effects become important in modelling such processes, as the injectedsteam will eventually condense at sites in the reservoir. Hanamura and Ka-viany [33] describe such a situation, where the injected steam condenses at afront, which then propagates through the porous medium. Bruining et al [16]formulate a model problem for steam injection into hot porous rocks, neglect-ing capillary effects, but including the effects of phase change, at a knownphase change temperature. With the thermal and phase change effects, theassumptions behind the IMPES method are no longer valid, and alternativecomputational strategies must be devised. Furthermore, the modelling oftwo-phase flow and phase change in porous media on smaller scales is often


dominated by capillary effects, which alter the structure of the mathemat-ical model. In particular, where capillary effects are important, saturationequations are often of degenerate parabolic type.

The development of Proton Exchange Membrane (PEM) fuel cells is atechnological advance prompted by environmental concerns. Fuel cell tech-nology has recently received much interest as the automotive industry hasrecognized the need for low emission power supplies as an alternative to theinternal combustion engine. As with steam injection processes in oil recov-ery, condensation of steam is also observed to occur in the porous electrodesof Proton Exchange Membrane (PEM) fuel cells, and has received attentionfrom a number of authors, for example, [12, 21, 56, 58, 70]. Consider the

Hydrogen Hydrogen

Coolant

Coolant

Anode

Cathode

Porous

electrodes

Gas

channels

Graphite

plates

Catalyst +

Membrane

Oxygen Oxygen

Figure 1.3: A PEM fuel cell.

simplified fuel cell configuration illustrated in Figure 1.3. The two electrodesindicated consist of a porous material. Hydrogen and oxygen diffuse throughthe electrodes to the membrane, where they react, producing water vapour,which is then transported back through the cathode. Condensation of thiswater vapour is observed to occur at certain locations within the cell, result-ing in liquid water within the cathode, or emerging into the oxygen channel.The accumulation of liquid water in the gas flow channels impairs the efficientdelivery of reactant gas to the electrode, while flooding of a porous electrodeinhibits the diffusion of gas through the medium, thereby reducing the supplyof reactant to the membrane. However, the membrane must remain hydratedfor the reaction to occur, and the topic of “water management” draws on the


effects of water in many inter-related processes inside the cell. The effects ofwater condensation can clearly be detrimental to the efficiency of operationof the fuel cell, and as such, a mathematical investigation of phase changein a porous medium is of interest to fuel cell manufacturers. In particular, amodel problem amenable to computational study is desired.

Models of two-phase flow with phase change in porous fuel cell electrodesare seen to include thermal and capillary effects, resulting in degenerateparabolic equations as described above. A number of recent studies haveused simplifications and regularisations of the model problems in order todeal with the numerical difficulties presented by their singular, degeneratenature. Bradean et al [12] identify regions of water vapour oversaturationwithin a dry fuel cell electrode, where condensation is likely to occur, butdo not include phase change effects in their model. He et al [35] presenta steady-state model problem, in which the liquid saturation dependence isremoved from the capillary pressure and relative permeability terms. There-fore, the degeneracy and singularity in the problem as the liquid saturationvanishes is avoided. In [70], another steady-state model problem is described,for two-phase, multicomponent flow in porous electrodes, but neglecting ther-mal variations and phase change effects. Natarajan and Nguyen [51] solvenumerically a time-dependent problem which includes phase-change effects.Saturation dependence in the relative permeability is included in their model,but regularised in order to simplify the computation. The same regularisa-tion is used by Mazumder and Cole [50]. The effect of such regularisations is,in general, to smear out sharp interfaces. A time-dependent problem whichmakes no mention of any such regularisation, but which assumes isother-mal conditions and does not include phase change, appears in [56]. A morerecent study by Birgersson et al [10] considers the steady-state flow andphase-change with no apparent artificial regularisation added to the prob-lem. In this thesis, we are primarily concerned with model problems forphase change in porous media which encompass all the mathematical diffi-culty of those appearing in the reservoir and fuel cell literature, but with aview to developing methods which may be applied in quite general, ratherthan specific, settings. With this in mind, we note here that we shall onlyconsider single component, two-phase flow, rather than the multicomponentflows described in the fuel cell literature, where oxygen, nitrogen and watermay all coexist in the porous media. Multiphase, multicomponent compu-tations will, in general, be more expensive than the computations which wedescribe here. However, the issues of singularity and degeneracy which we


tackle here will not be further complicated by adding extra components toour model.

More theoretical and general studies have been presented by other au-thors. A steady-state, one-dimensional study by Udell [64] investigates theeffects on a sand pack, which contains water, of heating the layer at thetop and cooling the layer from below. Experimental results indicate that,at steady state, there may be three distinct zones within the porous pack:a vapour zone at the top, a liquid zone at the bottom, and a two-phasezone in between. The basic set up is shown in Figure 1.4. In the two-phasezone there is a counterflow of liquid, driven upwards by capillary forces, andvapour, driven downwards by a pressure gradient. Udell presents a modelof the two-phase zone that assumes a constant temperature throughout thiszone, with condensation and evaporation occurring at the lower and upperinterfaces, respectively. The vapour in the two-phase zone is assumed to befully saturated. The model problem is solved to give a saturation profile,which indicates the length of the two phase zone. A similar study is per-formed by Torrance [63], with heating from the bottom, and similar resultsare obtained.

heat

cool

vapour

two−phase

liquid

z

z=D

z=L

z=0

Figure 1.4: A three-zone system for Udell’s experiment.

The isothermal two-phase assumption made by Udell [64] appears to bepopular throughout the literature. Temperature variation throughout a two-phase zone may be critical in applications such as fuel cell design. A recentstudy by Wang and Wang [71] specifically examines the fuel cell setting


under nonisothermal conditions, and shows numerical results for a two-phasezone. Also, a more complete steady-state model for Udell’s problem [64] ispresented in [14]. Temperature effects are included in this model, and phasechange is allowed to occur throughout the two-phase zone. A numericalmethod is developed for locating interfaces between the single-phase andtwo-phase regions, and is described algorithmically in [15].

Now we consider the time-dependent Udell problem. Specifically, we con-sider the problem of locating the interface between a single-phase region anda two-phase region in a single component, two-phase mixture, before thesystem has reached a steady-state. Such problems are of interest to fuel cellmanufacturers when considering the effects of start-up and shut-down of cells.For the numerical solution of such a problem, fixed-domain, front capturingmethods appeal. A method, based on mixture quantities, and using the samesaturation assumption as Udell, is described by Wang and Beckermann [68],and implemented in [66]. Another model which has been used in the fuel cellliterature penalizes any vapour not at saturation pressure into condensing ata large rate (see, for example, [21, 50]). Both of these methods require thesolution of a fixed-domain problem, from which the interface location canbe recovered. Computations using these methods have often been performedunder the various simplifying assumptions which we have described above.Due to these assumptions, and the lack of analytical solutions to the cou-pled model problems, the challenge remains to show, either analytically ornumerically, that such a computational capturing method for nonisothermal,two-phase flow with phase change in porous media yields an accurate solu-tion in time. We aim to formulate model problems and develop capturingmethods for their solution, for which we can show that the interface evolveswith the correct location and velocity.

1.4 Thesis overview

In this thesis, we formulate mathematical models for phase change in porousmedia, and related model problems, and develop numerical methods for thesolution of these problems, both steady-state and time-dependent. Through-out the thesis, we highlight points of particular mathematical interest. Theseare intended simply to aid the motivation for and understanding of our work,as well as to suggest directions of further interest. Most of our mathematicaldiscussion will be illustrated using one-dimensional model problems. The


emphasis throughout is on developing understandable, reproducible solutionmethods, rather than rigorous mathematical analysis.

The remainder of this thesis is organised as follows. In Chapter 2, wediscuss in some detail two prototype interface problems, which are relatedto the process of phase change in porous media. We describe formulationsof the Stefan problem of phase change, and the Porous Medium Equation,and discuss numerical capturing and tracking methods to approximate theirsolutions. A new asymptotic analysis of commonly used smoothing strate-gies is presented for a smoothed, steady-state Stefan problem. Also, carefulnumerical convergence studies are presented, in preparation for a comparisonwith the results of Chapter 4.

In Chapter 3, the steady-state, one-dimensional problem of phase changein porous media is described. This is an extension of an existing model, allow-ing for compressible vapour. Two numerical methods are described for thesolution of this free interface problem, and numerical results are presented,showing good agreement between the two methods. These results are usedas benchmark solutions for the time-dependent and two-dimensional compu-tations in Chapter 4.

In Chapter 4, the phase change in porous media problem is extended toinclude time-dependence, and reformulated as a fixed-domain problem. Anumerical capturing method is developed for the solution of this problem,avoiding several simplifications which have commonly appeared in the liter-ature. Computational solutions are shown to evolve to the correct steady-states predicted by the methods of Chapter 3. An analytical solution isfound for a reduced model problem, and numerical convergence studies us-ing this exact solution show that solutions from our capturing method areindeed convergent. Furthermore, the convergence rates are comparable withthe methods used for much simpler scalar problems, as described in Chap-ter 2. The implementation is extended to two dimensions, and computationalresults are shown.

In Chapter 5, we summarize our findings, describe ongoing work, andsuggest directions for future work.

16

Chapter 2

Solutions to prototype interfaceproblems

In this chapter, we study two prototype free and moving interface problemsof particular relevance, namely the two-phase Stefan problem and the PorousMedium Equation. We will formulate these problems mathematically, anddiscuss some analytical and numerical solution techniques for them. In par-ticular, one-dimensional model problems are used for illustration. Much ofthis chapter is a review, intended as a mathematical background to accom-pany and guide the applied work in later chapters. We also present a newasymptotic analysis of smoothing strategies applied to interface problems,and suggest further work.

2.1 The steady-state, two-phase Stefan

problem

2.1.1 The one-dimensional problem

First of all, consider the steady-state, one-dimensional ice/water problemwhich we briefly discussed in Chapter 1 (see (1.1), ff.). We suppose that iceoccupies a region 0 < z < s, and that liquid water occupies a region s <z < D. At z = s, there is a melting/freezing interface which separatesthe two regions. In each of the two distinct regions, the temperature isgoverned by a steady-state heat equation. At z = 0, suppose we have a giventemperature, T0, which is below the melting temperature. At z = D, supposewe have a given temperature, T1, which is above the melting temperature.Also, take the melting temperature to be zero (This is for ease of illustration,and temperature is measured in degrees Celsius in the physical problems inthis chapter. In later chapters, where phase change temperatures are notfixed, and the ideal gas law is important, we revert to temperature measured

Chapter 2. Solutions to prototype interface problems 17

in Kelvins). Then a mathematical model for the temperature throughout thesystem consists of two boundary value problems in each of the two regions,as follows:

(KiceTz)z = 0 0 < z < s,T (0) = T0 (< 0),T (s−) = 0,

(KwaterTz)z = 0 s < z < D,T (s+) = 0,T (D) = T1 (> 0).

(2.1)

Now, since the interface position s is an unknown in the problem, we requireone more condition. Energy should be conserved across the interface. Inother words, the heat flux should be continuous across the interface, giving

−[KTz

]s+s−

= 0. (2.2)

It is a straightforward matter to solve the problem by calculating s usingthe three interface conditions, and then finding the temperature T (z) inthe ice and the liquid water regions. Let us also consider an alternativeformulation, to show how we may “capture” the interface from the solutionof a transformed problem.

We can reformulate the disjoint domain problem (2.1)-(2.2) as a linearproblem for a transformed variable over the domain [0, D]. Let

v(T ) =

∫ T

T0

K(ξ) dξ, (2.3)

where the conductivity K is given by

K(T ) =

Kice T < 0

Kwater T > 0,

or equivalentlyK(T ) = Kice + (Kwater − Kice)H(T ),

where H is the Heaviside function. In the context of the time-dependentproblem, which we shall discuss in the next section, (2.3) is known as theKirchoff transformation. Then (2.1)-(2.2) becomes

(v(T ))zz = 0,

v|z=0 = 0, v|z=D = v(T1). (2.4)


Trivially, this has solution

v(T (z)) =v(T1)

Dz, (2.5)

and the temperature T (z) can then be recovered using (2.3). The interfacez = s is then “captured” using the condition that T (s) = 0. For the steady-state problem here, an exact solution is available. The steady-state position,s, of the freezing/melting interface is given by

s =

(T0Kice

T0Kice − T1Kwater

)D. (2.6)

Then the temperature profile is given by

T (z) =

−T0

sz + T0 0 ≤ z ≤ s,

T1

D−s(z −D) + T1 s ≤ z ≤ D.(2.7)

It appears that the discontinuity in the conductivity function K at the in-terface does not hinder us analytically, or indeed, numerically.

2.1.2 The two-dimensional problem

Now let us consider how the problem is solved in two dimensions.Consider the problem shown in Figure 2.1. Liquid water and ice occupy

regions Ω1,Ω2 respectively, and are separated by a freezing/melting inter-face Γ. We now simply extend the mathematical model (2.1)-(2.2) to twodimensions, giving

∇.(Kice∇T ) = 0 (x, y) ∈ Ω2

T (x, 0) = T0(x) (< 0),T = 0 on Γ−,

∇.(Kwater∇T ) = 0 (x, y) ∈ Ω1,T = 0 on Γ+,

T (x,D2) = T1(x) (> 0),T (0, y) = Tleft(y), T (D2, y) = Tright(y),

(2.8)

together with the heat balance across the interface:

−[(K∇T ).n

]12

= 0. (2.9)


ICE

WATERΩ1

Ω2

00

Γ

D1

D2T = T1

x

y

T = T0

Figure 2.1: The steady-state, two-phase Stefan problem in two dimensions.

Here, n is the unit normal to the interface Γ. Once again, we may solve forthe temperature and interface location by a change of variables which leavesa fixed-domain problem. Making the change of variables (2.3)

v(T ) =

∫ T

Tref

K(ξ) dξ,

for a chosen reference temperature Tref , the problem reduces to

v = 0, (x, y) ∈ (0, D1) × (0, D2)v(x, 0) = v(T0(x)),

v(x,D2) = v(T1(x)),v(0, y) = v(Tleft(y)),

v(D1, y) = v(Tright(y)).

(2.10)

That is, we may solve for the interface location and the temperature in thedisjoint regions by first solving Laplace’s equation for v on the fixed domain.Given the solution v, the temperature T is easily recovered from the following:

T =

v+KiceTref

Kiceif v < −KiceTref ,

v+KiceTref

Kwaterif v ≥ −KiceTref .

(2.11)


Analytical solutions to the problem (2.10) may be found using Fourier seriesmethods. For illustration here, consider a problem where we take T0 to beconstant, using Tref = T0, D1 = D2 = 1, take T1(x) to be 1-periodic, andgive periodic conditions in the x-direction. Then the solution v of the fixeddomain problem is given by

v(x, y) =a0

2y +

∞∑

n=1

(an

sinh 2nπcos 2nπx+

bnsinh 2nπ

sin 2nπx

)sinh 2nπy,

(2.12)where the Fourier coefficients are given by

a02

=∫ 1

0v(T1(x)) dx,

an = 2∫ 1

0v(T1(x)) cos 2nπx dx, n = 1, 2, ... ,

bn = 2∫ 1

0v(T1(x)) sin 2nπx dx, n = 1, 2, ... .

(2.13)

In Figure 2.2, we plot the steady-state solution to the problem (2.8), using

0

0.2

0.4

0.6

0.8

10

0.51

−2

−1

0

1

2

3

4

5

6

y

Temperature profiles

x

tem

pera

ture

T

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

Temperature contours

011

2 233

4 4

ICE

WATER

Figure 2.2: Temperature profile and contours for steady-state two-phase Ste-fan problem.

periodic conditions in x, with Kice = 2.2, Kwater = 0.55, T0 = −2 andT1(x) = 3 + 3 cos 2πx. This two-dimensional problem is simply a sinusoidal


perturbation to the one-dimensional problem (2.1), with T1 = 3, which,by (2.6), has the interface at s = 8/11 ≈ 0.73. This average interface positionis clear in Figure 2.2.

It is worth noting here that our solution method, namely solving thefixed-domain problem (2.10) for the transformed variable v, then recover-ing temperature using (2.11), is valid for more general problems than (2.8).Since v has continuous derivatives everywhere, the heat balance (2.9) willhold across any interface. The temperature recovery relies only on the valueof v, and does not require explicit knowledge of the location of any interfaces.Therefore, this method may be applied in general to problems where there areno interfaces, a single interface, or multiple interfaces. This idea extends tocapturing methods for time-dependent problems, which gives these methodsa particular appeal over front-tracking, as we shall see in the next section. InFigure 2.3, we plot the steady-state solution to a two-phase Stefan problemusing periodic conditions in x, with Kice = 2.2, Kwater = 0.55, T0 = −2 andT1(x) = 3 + 3.5 cos 2πx. Clearly, there are two interfaces, which separatethe liquid water from two regions containing ice - one near the cold intervalalong the upper boundary, and one near towards the lower boundary.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

Temperature contours with two interfaces

0

11

22

3 3 44

0

0

WATER

ICE

ICE

Figure 2.3: Temperature profile and contours for a steady-state two-phaseStefan problem with multiple interfaces.


2.1.3 A smoothing method and asymptotic results

The two-phase Stefan problem which we have described here has been illus-trated by examples for which exact solutions are available. Such analyticalsolutions are not generally available for problems on irregular domains, prob-lems with different boundary data, generalized Stefan problems with extraheat sources, for example, and for time dependent problems. In these cases,we inevitably resort to numerical solution methods. Any code which involvesthe step of temperature recovery from the transformed variable will require“if” statements, and discretizations may be hindered by the discontinuoustemperature gradients. One approach to take is to smooth the discontinuousconductivity function K(T ) over some radius in T , and solve the resultingsmoothed problem. While some implementations of the enthalpy methodfor the transient problem, which we will describe in the next section, oftenmake use of the exact conductivity, some degree of smoothing is often ap-plied in order to aid computations (see, for example, [3, 20]). It appears thatlittle analysis of such smoothing methods has been presented in the litera-ture. Here, we present a brief analysis of a smoothing strategy applied toour one-dimensional, steady-state Stefan problem.

We aim to give a regularization of the problem 2.1 by smoothing theconductivity function, leaving a smooth, fixed-domain problem. Let us write

Kε(T ) = Kice + (Kwater − Kice)Hε(T ), (2.14)

where Hε is a C∞ symmetric regularization of the Heaviside function, withsmoothing radius ε. Then the problem

v(T ) =∫ TT0Kε(ξ) dξ,

(v(T ))zz = 0,

v|z=0 = 0, v|z=D = v(T1)

(2.15)

has a C∞ solution T (z) which should tend to the exact solution of the Stefanproblem as ε → 0. We do not prove this here, but observe this numerically.Of particular interest is the difference in the location of the interface whereT = 0 between the exact and smooth solutions, and the dependence of thiserror on ε. We see from (2.5) that the interface location z = s is given by

s =v(0)

v(T1)D. (2.16)


Since the smooth solution satisfies

v(T ) = Kice(T − T0) + (Kwater − Kice)

∫ T

T0

Hε(ξ) dξ, (2.17)

we see that the value of s for this smooth solution can be found approximatelyby considering the asymptotic evaluation of the integrals

∫ 0

T0

Hε(ξ) dξ, and

∫ T1

T0

Hε(ξ) dξ,

with T0 < 0, T1 > 0 and ε ≪ |T0|, T1, for our choice of smoothed Heavi-

side Hε. We illustrate this for two common choices in the examples below.

Example 2.1

Let

Hε(X) =1

2+

1

πtan−1

(X

ε

). (2.18)

Then ∫ T

T0

Hε(ξ) dξ =1

2(T − T0) +

1

πI(T ), (2.19)

where

I(T ) = T tan−1 T

ε− T0 tan−1 T0

ε− ε

2log

(T 2 + ε2

T 20 + ε2

). (2.20)

Recalling the identity

tan−1 1

X+ tan−1X =

−π/2 X < 0π/2 X > 0

,

and the expansions

tan−1X = X − X3

3+ ..., log(1 +X) = 1 − X2

2+ ...,

we find

I(0) ∼ π

2T0+(1+log |T0|)ε−ε log ε, I(T1) ∼

π

2(T1+T0)−

(log

T1

|T0|

)ε.

(2.21)


Hence, ∫ 0

T0

Hε(ξ) dξ ∼ 1

π[(1 + log |T0|)ε− ε log ε] ,

and ∫ T1

T0

Hε(ξ) dξ ∼ T1 −(

1

πlog

T1

|T0|

)ε.

So we havev(0) ∼ −KiceT0 + E(Fε− ε log ε),

where

E =Kwater − Kice

πand F = 1 + log |T0|,

and also

v(T1) ∼ A− E logT1

|T0|ε,

whereA = KwaterT1 − KiceT0.

Finally, substituting into (2.16), recalling the exact interface location (2.6),

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−3

−2

−1

0

1

2

3

4

5

6

7

Smooth approximations as ε −> 0, with T0=−3 T1=7 kice

=0.8 kwater

=0.25 N=80

smooth solutions approaching exact solution (dots)as ε −> 0

Figure 2.4: Smooth approximations converging to exact, nonsmooth steady-state temperature (dots) as ε→ 0, for Example 2.1.


and noting that 1/v(T1) ∼ 1A

(1 + E

Alog T1

|T0|ε), we find the interface location

to the smoothed problem is given by

s ∼ sexact +E

A

[(F − KiceT0

Alog

T1

|T0|

)ε− ε log ε

]D. (2.22)

Clearly, s → sexact as ε → 0. In Figure 2.4, we show the convergence ofthe smooth approximations to the nonsmooth steady-state temperature. InFigure 2.5, we show a close-up of the computed smooth solution, showinggood agreement with the asymptotic interface location (2.22).

0.108 0.11 0.112 0.114 0.116 0.118

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

ε=0.01 with T0=−3 T1=7 kice

=0.8 kwater

=0.25 N=400

z

T

exact solution computed smooth solutionasymptotic interface position

Figure 2.5: Computed smooth approximation and asymptotic form, for Ex-ample 2.1.

Example 2.2

Let

Hε(X) =1

2

1 + tanh

(X

ε

). (2.23)

Here, we use the fact that

log

(cosh

X

ε

)∼ |X|

ε− log 2 for ε≪ |X|,


to obtain∫ 0

T0

Hε(ξ) dξ ∼ log 2

2ε and

∫ T1

T0

Hε(ξ) dξ ∼ T1 + o(ε, ε log ε).

Proceeding as in Example 1, we find that

s ∼ sexact +

(Kwater − Kice

KwaterT1 − KiceT0

)log 2

2D ε. (2.24)

We notice the order of error in each of these examples, and conjecture thatwe cannot achieve a higher order error if our choice of smoothed Heavisideis monotonic increasing. The reason is due to the fact that the integral

∫ 0

T0

Hε(ξ) dξ = O(ε),

for any monotonic increasing Hε, as suggested by Figure 2.6.

0

1

0.5

O(ε)

O(1)

X

Hε(X)

Figure 2.6: A sketch of a monotonic increasing smoothed Heaviside function,Hε. Clearly,

∫ 0

XHε(ξ) dξ = O(ε), for X < 0.

The smoothing method presented here can, of course, be applied to higherdimensional problems, and time-dependent problems. We suggest further in-vestigation of smoothing methods and a more formal analysis of our techniquefor the steady-state Stefan problem as open problems.


2.1.4 The method of residual velocities

Recent work by Donaldson [22] presents a trial method for solving free inter-face problems, with a view to solving more complex problems for which fixeddomain methods may not be straightforward to formulate or implement. Forillustration of the method, particular attention is paid to the steady-stateStefan problem, in two dimensions. Here, we describe the method for theone-dimensional problem. Consider again the problem (2.1)-(2.2), as shownin Figure 2.7. We can think of the problem as consisting of two boundaryvalue problems, one on either side of the interface z = s. If the value of s isknown, then we require only two conditions at the interface, but since s isan unknown in the problem, we require three conditions there.

ICE WATER

0z

Ds

T = Tbot T = Ttop

(KiceTz)z = 0 (KwaterTz)z = 0

[KTz

]+−

= 0 (i)

T+ = 0 (ii)

T− = 0 (iii)

Figure 2.7: The steady-state, one-dimensional Stefan problem (2.1)-(2.2).

The idea of the Residual Velocity method described in [22] is to makean initial guess for s, then to solve the boundary value problems in the iceand water regions, using two of the three interface conditions. The thirdcondition, in general, will not be satisfied, unless the interface is at the exactsteady-state location. An interface “velocity” is defined to be equal to thecomputed residual in this third condition, and the interface is subsequentlymoved according to this velocity and the chosen time-stepping scheme. Thisprocess is repeated until the interface approaches some limit for large “time”.This is essentially a front-tracking method applied to steady-state problemson either side of the front. While the solution method treats the residualsas velocities, the evolution of the interface in “time” is completely artificial,and only the steady-state is meaningful.


For the one-dimensional problem in Figure 2.7, we now demonstrate theResidual Velocity method using each of the three conditions as velocities.Suppose the initial guess for the interface is s0.

Example 2.3

Using the residual in condition (i) as the velocity (ie. solving the boundaryvalue problems using conditions (ii) and (iii)), the interface velocity is givenby

ds

dt= −

(Kwater

TtopD − s

+ KiceTbots

), s(0) = s0. (i)

Example 2.4

Using the residual in condition (ii) as the velocity (ie. solving the boundaryvalue problems using conditions (i) and (iii)), the interface velocity is givenby

ds

dt= −

(Ttop −

KiceTbot

Kwater

s−D

s

), s(0) = s0. (ii)

Example 2.5

Using the residual in condition (iii) as the velocity (ie. solving the boundaryvalue problems using conditions (i) and (ii)), the interface velocity is givenby

ds

dt= −

(Tbot −

KwaterTtop

Kice

s

D − s

), s(0) = s0. (iii)

In Figure 2.8, we plot solutions of the ODE’s from Examples 2.3-2.5. Wetake D = 1, Tbot = −1, Ttop = 1, Kice = 2.2, Kwater = 0.55, and computesolutions given two different initial conditions, s0 = 0.5, and s0 = 0.97. Inboth cases, we see that all three velocity choices give an interface whichevolves and converges to the correct steady-state solution s = 0.8.

For the one-dimensional problem considered here, the analytical solu-tion of the boundary value problems is trivial. In higher dimensions, wherethe interface geometry must be taken into account, solution of these prob-lems requires numerical methods. Also, the choice of residual velocity is not


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Steady−state interface location, using Residual Velocity method

"time" t

inte

rfac

e lo

catio

n, s

residual velocity (i)residual velocity (ii)residual velocity (iii)

Figure 2.8: Convergence to steady-state interface location, using ResidualVelocity computations.

limited to the three shown here. The interface conditions used in the compu-tations may be constructed using linear combinations of the three physicalconditions (i)-(iii), and the interface should evolve to the same steady-state.In [22], a discretization of the interface for a two-dimensional problem is de-scribed, and the numerical properties of various choices of velocity appliedwith certain time-stepping schemes are analyzed.

2.2 The time-dependent, two-phase Stefan

problem

In this section, we discuss the extension of the model two-phase Stefan prob-lem to include time-dependence. With a view to solving the phase change


problems in porous media described in later chapters, we look in particularat the formulation of the problem, exact solutions, and the capturing meth-ods which have been developed for numerical solution. The contents of thissection are largely drawn from well established work, which is particularlywell described in detail by Crank [20] and Alexiades & Solomon [3].

2.2.1 Mathematical formulation

Let us consider the extension of the problem (2.1) to the time-dependentproblem. Conceptually, we imagine a partially melted block of ice, with amoving freezing front, before a steady-state has been reached. Mathemati-cally, we have

ρciceTt = (KiceTz)z 0 < z < s(t),T (0) = Tbot (< 0),T (s−) = 0,

ρcwaterTt = (KwaterTz)z s(t) < z < D,T (s+) = 0,T (D) = Ttop (> 0).

(2.25)

Here, ρ represents density, which we assume is the same in both liquid andsolid phase, and cice,water are the specific heat capacities of ice and liquidwater respectively. Once again, we require an extra condition at the interface,which represents conservation of energy across the interface. Consider thediagram shown in Figure 2.9.

Suppose that the interface is moving to the right, so that it is a freezingfront. In small time δt, the front moves a small distance δz. During thistime interval, the heat which flows out (per unit area) into the ice region isapproximately KiceTz δt. This must equal the heat which flows in from thewater region, plus the heat released upon freezing the mass (per unit area)ρδz. Thus we have

(KTz

)iceδt ≈

(KTz

)water

δt+ Lρδz,

where L is the specific latent heat of freezing. Taking the limit δz, δt → 0,we find

ρLs(t) = −[KTz

]waterice

, (2.26)


δz, δt

s(t)ICE

WATER

Figure 2.9: Heat balance at the freezing interface - the Stefan condition.

where s is the interface velocity. This condition for the interface velocity isgenerally referred to as the Stefan condition. It is worth noting that wewill arrive at this same condition regardless of whether we consider a meltingor freezing process.

In order to complete the specification of the time-dependent problem, wealso add an initial temperature distribution, and an initial interface location.That is, we also give

T (z, 0) = Tinit(z), 0 < z < D, s(0) = s0. (2.27)

Now, defining

αice =Kice

ρcice, αwater =

Kwater

ρcwater,

our two-phase Stefan problem becomes

Tt = αiceTzz 0 < z < s(t),T (0) = Tbot (< 0),T (s−) = 0,

Tt = αwaterTzz s(t) < z < D,T (s+) = 0,T (D) = Ttop (> 0),

ρLs(t) = −[KTz

]waterice

,

T (z, 0) = Tinit(z), 0 < z < D,s(0) = s0.

(2.28)


2.2.2 Analytical solutions

Exact, analytical solutions to two-phase Stefan problems such as (2.28) arenot generally available. The two most common types of solution, namely theNeumann similarity solution and the travelling wave solution, appear to havelimited application in real world problems, but such solutions are valuable ingiving mathematical insights into the Stefan problem, providing useful ap-proximations, and evaluating the performance of numerical schemes. Here,with a view to constructing analytical solutions to the moving interface prob-lems which we will encounter in later chapters, we describe both a Neumannand travelling wave solution.

The Neumann solution

A Neumann similarity solution is available for a problem on a semi-infinitedomain, with an initial condition of all liquid water or all ice, thus with afreezing or melting interface moving from one throughout the domain, start-ing at the fixed boundary. Here, we consider the problem (2.28), with D = ∞and Tinit = Ttop. That is, we have a freezing problem, with the freezing in-terface moving through the domain from z = 0. We follow similar analysesfor melting problems by Alexiades and Solomon [3], and Zwillinger [75].

Guided by the heat equations either side of the interface, we seek solutionsfor t > 0 of the form

T (z, t) = f(η) in ice, T (z, t) = g(η) in water, where η =z√t. (2.29)

For such solutions to exist, we require

s(t) = β√t, (2.30)

for some constant β to be determined as part of the solution. The Stefanproblem (2.28) then becomes

αicef′′ + 1

2ηf ′ = 0 0 < η < β,

f(0) = Tbot, f(β) = 0,

αwaterg′′ + 1

2ηg′ = 0 β < η <∞,

g(β) = 0, g(∞) = Ttop,

12ρLβ = Kicef

′(β) − Kwaterg′(β).

(2.31)


The solution of this system is given by

f(η) = Tbot

(1 −

erf( η2√αice

)

erf( β2√αice

)

)0 < η < β, (2.32)

and

g(η) = Ttop

(1 −

erfc( η2√αwater

)

erfc( β2√αwater

)

)β < η <∞. (2.33)

where the functions erf and erfc are the error function and complementaryerror function respectively. The value of β, which gives the interface location,satisfies the equation

−√π

2ρLβ =

KwaterTtop√αwater

e− β2

4αwater

erfc( β2√αwater

)+KiceTbot√

αice

e− β2

4αice

erf( β2√αice

). (2.34)

In order to construct the Neumann solution given above, we must firstsolve (2.34) for β. This may be achieved using a Newton iteration, or bi-section method, for example. A typical solution is shown in Figure 2.10.

Travelling wave solutions

When seeking an analytical solution to a two-phase Stefan problem, it issometimes more straightforward to construct a travelling wave solution ratherthan a Neumann solution. A travelling wave solution is one whose profileremains unchanged in a reference frame moving with constant speed.

For the following problem,

Tt = αiceTzz z < s(t),T (s−) = 0,

Tt = αwaterTzz z > s(t),T (s+) = 0,

ρLs(t) = −[KTz

]waterice

,

(2.35)

we seek solutions of the form

T = F1(ξ) for ξ < 0 (ice), T = F2(ξ) for ξ > 0 (water), ξ = z − ct.(2.36)


0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10−3

−10

−5

0

5

Neumann freezing problem, ρ=1000 L=3.3e+005 cwater

=4200 cice

=2000 Kwater

=0.55 Kice

=2.2 Ttop

=5 Tbot

=−10 .... β=0.000342

η

tem

pera

ture

g(η)

f(η)

ICE

WATER

Figure 2.10: A typical Neumann solution to the freezing problem (2.32)-(2.34).

Here, c is a constant, which is the speed of the interface. So s(t) = ct, andfor c > 0, we have an interface moving to the right. We have deliberatelynot specified boundary or far-field conditions. Now, with parabolic problemseither side of an interface on which we give three conditions, we expect togenerate a one-parameter family of solutions. The system becomes

−cF ′1 = αiceF

′′1 ξ < 0,

F1(0) = 0,

−cF ′2 = αwaterF

′′2 ξ > 0,

F2(0) = 0,

ρL = Kice

αiceA1 − Kwater

αwaterA2,

(2.37)

which has solution

F1(ξ) = A1(1 − e− c

αiceξ) ξ < 0,

F2(ξ) = A2(1 − e− c

αwaterξ) ξ > 0,

(2.38)


where

A1 =αice

Kice

(ρL+

Kwater

αwaterA2

). (2.39)

We indeed have a one-parameter family of solutions; that is, a family of solu-tions parameterized by A2. Such travelling wave solutions can be constructedon finite domains by specifying the appropriate boundary conditions. On aninfinite or semi-infinite domain, notice that F2(ξ) → A2 as ξ → ∞, andso A2 is seen to be the far-field temperature. However, F1(ξ) is exponen-tially large as ξ → −∞, and this we note again that this analytical solutionserves mainly as a mathematical tool rather than having any physical mean-ing. Furzeland [31], for example, uses exact solutions such as those presentedhere for evaluating the performance of numerical methods applied to Stefanproblems. A typical profile is shown in Figure 2.11.

−3 −2 −1 0 1 2 3 4 5 6 7

x 10−7

−60

−50

−40

−30

−20

−10

0

10

ξ

Travelling wave profile with c=1, A2=3

tem

pera

ture

F1(ξ)

F2(ξ)

ICE

WATER

Figure 2.11: A typical travelling wave solution to the freezing problem (2.37).


2.2.3 Formulation of the enthalpy method

In most cases requiring the solution of a two-phase Stefan problem, a numer-ical solution is sought. Let us consider strategies for the numerical solutionof the problem 2.1. One option, as discussed in Chapter 1, is to employ afront tracking method. This is a method which explicitly moves the interfaceat each time step. We consider a finite difference method which updatesthe temperature, and moves the interface between two time levels, which wedenote n and n+ 1, and outline this method in Algorithm 2.1.

Algorithm 2.1 (Front tracking)Given an interface location sn and temperature at a time level n.

1. Discretize the interval 0 < z < sn, and solve the heat equation in theice, subject to T = 0 at the interface.

2. Discretize the interval sn < z < D, and solve the heat equation in thewater, subject to T = 0 at the interface.

3. Use the Stefan condition (2.26) to calculate the velocity s.

4. Move the interface according to sn+1 = sn + ks.

5. Go back to step 1.

Here, k is the size of the time step between time levels n and n + 1. No-tice that with each time step, remeshing takes place in steps 1 and 2. Auniform grid on (0, D) will not suffice for front-tracking, since the interfacewill, in general, not move by exactly one grid point with each time step. Analternative is to reselect the size of the time step each time the interface ismoved, in order that it will move by exactly one grid point. Details of suchapproaches to front-tracking are described by Crank [20]. Clearly, in termsof ease of implementation, a front-capturing method is preferable. Furtheradvantages arise in higher dimensional problems, where tracking methodswould encounter additional complication due to the interface geometry.

Now we consider how we may reformulate the model problem (2.25) overthe fixed domain (0, D). Since the heat equation in each of the disjoint iceand water regions describes energy conservation, we aim to write an equationfor energy conservation throughout the entire domain. A measure of energy


over the fixed domain must take into account the specific latent heat of phasechange. We define the enthalpy (per unit volume), E, by

E(T ) =

ρciceT for T < 0 (ice),∈ [0, ρL] for T = 0 (“mushy′′),ρcwaterT + ρL for T > 0 (water).

(2.40)

At a temperature T = 0, the ice or water may be undergoing phase change, sowe refer to it as “mushy”. Notice the jump in the enthalpy function at T =0, as shown in Figure 2.12. To deal with the discontinuous conductivity

−100 −80 −60 −40 −20 0 20 40 60 80 100−2

−1

0

1

2

3

4

5

6

7

8x 10

8

temperature T

enth

alpy

E

Enthalpy versus temperature

Figure 2.12: Enthalpy as a function of temperature.

function, we once again make the familiar transformation

v =

∫ T

0

K(ξ) dξ, (2.41)

which, in this context, is known as the Kirchoff transformation; the vari-able v is often referred to as the Kirchoff temperature. Then the interfaceproblem has now been reduced to the following fixed-domain problem for theenthalpy E:

Et = vzz 0 < z < D, t > 0E(0, t) = ρciceTbot,E(D, t) = ρcwaterTtop + ρL,E(z, 0) = Einit(z).

(2.42)


The challenge now is to devise an algorithm for solving (2.42) numerically,which carefully implements mappings between E, v and T . After giving aninitial temperature, we will step in time to solve (2.42), and then the locationof the interface will be recovered from the solution E (or v).

Let us consider a finite difference method. Suppose we have a grid zj =(j − 1)h for j = 1, ..., N + 1, where h = D/N . Let V n

j be the approximatevalue of v((j − 1)h, nk), where k is the time step to be used in the finitedifference scheme. Then an algorithm for the solution of (2.42) is given belowin Algorithm 2.2. Firstly, let us write explicitly the relationship between Eand v:

E(v) =

ρcice

Kicev v < 0,

ρcwater

Kwaterv + ρL v > 0,

∈ [0, ρL] v = 0;

(2.43)

v(E) =

Kice

ρciceE E < 0,

Kwater

ρcwater(E − ρL) E > ρL,

0 0 ≤ E ≤ ρL .

(2.44)

Algorithm 2.2 (Enthalpy method)Given V n:

1. Compute En using (2.43).

2. Update Ej for j = 2, ..., N by using finite differences for (2.42). ObtainEn+1j .

3. Recover V n+1j using (2.44).

The algorithm shown solves (2.42). In order to solve the original problem fortemperature and interface position, we can simply recover T from the solutionV , and then interpolate to find the interface location s(t). It remains todecide on the scheme to use for Stage 2 of the algorithm. An explicit schemeis easy to implement as follows

En+1j = En

j + µ(V nj−1 − 2V n

j + V nj+1

), (2.45)


for j = 2, ..., N , where µ = kh2 . However, we have the time step restriction

k ≤ h2 ρ

2min

(cice

Kice

,cwater

Kwater

). (2.46)

In situations where this is too restrictive , we may prefer an implicit scheme.However, an implicit scheme will require some clever way of deciding whichenthalpy range each grid point should be in at the next time step. We nowdescribe two ways of achieving this.

Nonlinear Gauss-Seidel iteration

Consider the PDEEt = vzz,

where

E(v) =

v/Kice v < 0∈ (0, ρL) v = 0

v/Kwater + ρL v > 0

, (2.47)

and so

v(E) =

KiceE E < 00 E ∈ [0, ρL]

Kwater(E − ρL) E > ρL

. (2.48)

Suppose we seek an iterative solve for the implicit scheme

En+1j − µ

(vn+1j−1 − 2vn+1

j + vn+1j+1

)= En

j . (2.49)

A Gauss-Seidel solver will solve the jth equation for the jth unknown usingthe latest available values of all other unknowns. Let us rewrite (2.49) as

En+1j + 2µvn+1

j = Enj + µ

(vn+1j−1 + vn+1

j+1

). (2.50)

So if we solve for each j in order, we can iterate to find vn+1j using Gauss-

Seidel as follows:

Ep+1j + 2µvp+1

j = Enj + µ

(vp+1j−1 + vpj+1

), (2.51)

at the pth iteration. Now the right hand side contains only known terms.However, we still have the nonlinear dependence E(v) given by (2.47). Nowlet

ψpj = Enj + µ

(vp+1j−1 + vpj+1

). (2.52)


So ψpj is known at the jth equation of the pth iteration for vn+1j . So (2.51) is

justEp+1j + 2µvp+1

j = ψpj . (2.53)

Now examine the three possibilities for vp+1j : ice, mushy or water.

Ice

vp+1j ≤ 0 ⇒ Ep+1

j = vp+1j /Kice ≤ 0,

and then (2.53) gives

vp+1j /Kice + 2µvp+1

j = ψpj ≤ 0, (2.54)

and hence

vp+1j =

ψpj

2µ+ 1/Kice

. (2.55)

Mushy

vp+1j = 0 ⇒ 0 < Ep+1

j < ρL,

and then (2.53) gives0 < ψpj = Ep+1

j < ρL. (2.56)

Water

vp+1j ≥ 0 ⇒ Ep+1

j = vp+1j /Kwater + ρL,

and then (2.53) gives

ψj,p = vp+1j /Kwater + ρL+ 2µvp+1

j ≥ ρL, (2.57)

and hence

vp+1j =

ψpj − ρL

2µ+ 1/Kwater

. (2.58)

Thus, the sign of and size of ψpj determines vp+1j . That is

vp+1j =

ψpj

2µ+1/Kiceψpj < 0

0 ψpj ∈ [0, ρL]

ψpj −ρL

2µ+1/Kwaterψpj > ρL

. (2.59)


This Gauss-Seidel idea is widely used, since it avoids the need for computingderivatives. The convergence of such a scheme may be accelerated usingsuccessive over-relaxation (see [3, 20], for example).

Smoothing methods and Newton iteration

An alternative to the trial-and-error methods for temperature recovery is tomake use of smoothed functions again. That is, we replace the discontinuousfunctions E and v from (2.40) and (2.41) with

Eε(T ) = (1 − Hε(T ))ρciceT + Hε(T )(ρcwater + ρL), (2.60)

andvσ(T ) = (1 − Hσ(T ))KiceT + Hσ(T )Kwater(T ), (2.61)

where H is a smoothed Heaviside, such as those in Examples 2.1 and 2.2,and ε, σ are smoothing radii in temperature. We then solve (2.42), for E,and the temperature recovery may then be achieved using a Newton iter-ation. This idea is particularly helpful for implicit time-stepping schemes,where Newton methods are commonly employed. In Figure 2.13, we plot asmoothed enthalpy function, taking ε = 5, and using the smoothed Heavisidefrom Example 2.2.

−100 −80 −60 −40 −20 0 20 40 60 80 100−2

−1

0

1

2

3

4

5

6

7

8x 10

8

temperature T

enth

alpy

E

Enthalpy versus temperature

exact enthalpy E(T)smoothed enthalpy

Figure 2.13: Enthalpy and smoothed enthalpy as functions of temperature.


We leave the investigation of the effect of such smoothing strategies onsolutions to the enthalpy formulation of the Stefan problem as an open prob-lem.

2.2.4 Mathematical justification for the enthalpy

method

Here we follow Alexiades [3] and Crank [20] to show that the Stefan conditionat the interface is satisfied by the weak solution to the enthalpy problem

Et = vzz z ∈ (0, D), t ∈ (0, τ),v(0, t) = v0, v(D, t) = v1,E|t=0 = E(Tinit).

(2.62)

Here, if T is temperature, then E(T ) is the enthalpy, and v(T ) is the Kirchoffvariable. Now consider the diagram shown in Figure 2.14.

C1

C1

C1

C1

C2

C2

C2

C2v = v0 v = v1

G1

G2

Γ : z = s(t)

τ

φ = 0 φ = 0

φ = 0

φ = φ(z, 0)0

0

t

Dz

Figure 2.14: The (z, t)-plane for the enthalpy formulation.

The curve Γ has z = s(t) being single-valued, and T = 0 with T continu-ous across the interface. First consider the whole region G = (0, D)× (0, τ),


and test functions φ(z, t) ∈ C∞(G), with φ(0, t) = φ(D, t) = φ(z, τ) = 0.Then we define a weak solution to (2.62) to be a pair of functions E, v whichsatisfy ∫ ∫

G

φEt − φvzz dz dt = 0,

Now, integrating by parts, or simply noting that

φEt = (φE)t −Eφt and φvzz = (φvz − vφz)z + vφzz, (2.63)

we see that∫ ∫

G

Eφt + vφzz dz dt =

∫ ∫

G

(φE)t − (φvz − vφz)z dz dt. (2.64)

Applying the boundary and initial conditions, we find∫ ∫

GEφt + vφzz dz dt = −

∫ D0φ(z, 0)E(Tinit) dz

+∫ τ0v1φz(D, t) − v0φz(0, t) dt,

(2.65)

and work with this as our weak formulation. It remains to show that a so-lution of (2.65) satisfies the Stefan condition on the moving interface. To dothis, we integrate in a similar fashion over the two domains G1, G2, shown inFigure 2.14. The curve Γ divides G into G1 and G2, which we consider to beice and water respectively.

In G1, we have∫ ∫

G1

Eφt + vφzz dz dt =

∫ ∫

G1

(φE)t − (φvz − vφz)z dz dt. (2.66)

Now, the integration this time requires use of Green’s Theorem in the Plane(or, for higher dimensions, the divergence theorem). Recall

∮

∂RP dx +Q dy =

∫ ∫

R

∂Q

∂x− ∂P

∂ydx dy,

where ∂R is positively oriented in the (x, y) plane. So the right hand sideof (2.66) is given by

RHS = −∮C1φE dz + (φvz − vφz) dt,

= −∫ s(0)0

φ(z, 0)E(Tinit) dz −∫ τ0v0φz(0, t) dt

−∫Γ−

(φE) dz + (φvz) dt,


since v = 0 on Γ. Thus we have, for G1,

∫ ∫G1Eφt + vφzz dz dt = −

∫ s(0)0

φ(z, 0)E(Tinit) dz

−∫ τ0v0φz(0, t) dt−

∫Γ−

(φE) dz + (φvz) dt.(2.67)

Now, for the water region G2. Arguing in the same way, we get

∫ ∫G2Eφt + vφzz dz dt = −

∫ Ds(0)

φ(z, 0)E(Tinit) dz

+∫ τ0v1φz(D, t) dt+

∫Γ+(φE) dz + (φvz) dt.

(2.68)Now, adding (2.67) and (2.68), and letting Γ−,Γ+ → Γ, we see that

∫ ∫G1∪G2

Eφt + vφzz dz dt = −∫ D0φ(z, 0)E(Tinit) dz

+∫ τ0v1φz(D, t) − v0φz(0, t) dt

+∫Γ

[φE]+− dz + [φvz]+− dt.

(2.69)

Now, comparing (2.65) and (2.69), we have

∫

Γ

[φE]+− dz + [φvz]+− dt = 0,

and hence, since φ and its derivatives are continuous across the interface,

∫

Γ

φ([E]+− dz + [vz]

+− dt

)= 0. (2.70)

Finally, since φ is an arbitrary test function, and since z = s(t) on Γ, wehave that

ds

dt= −

[vz]+−

[E]+−, (2.71)

which is precisely the Stefan condition.


2.2.5 Numerical results and convergence study

Now, we illustrate some interesting and important features of the enthalpysolution to the Stefan problem, which will help us to explain the behaviourof capturing methods for more complex problems. Firstly, we consider theproblem (2.28) with initially all liquid at T = 3, and subject to tempera-tures T = −2 at z = 0 and T = 3 at z = D = 1. Thus, a freezing frontpropagates through the domain. That is, our boundary and initial conditionsare

Tbot = −2, Ttop = 3, Tinit = Ttop, s(0) = 0.

The physical parameters we use are shown in Table 2.1.

Table 2.1: Constants for the Stefan problem

Symbol Interpretation Typical value Units (SI)

ρ density 1000 kg/m3

cwater specific heat capacity of water 4200 JK−1kg−1

cice specific heat capacity of ice 2000 JK−1kg−1

Kwater thermal conductivity of water 0.55 Wm−1K−1

Kice thermal conductivity of ice 2.2 Wm−1K−1

L specific latent heat of freezing 3.3 × 105 Jkg−1

For the results shown here, we have used Forward Euler time-stepping,with N = 20 grid points. In Figure 2.15, we plot a succession of temperatureprofiles in increasing time, together with the steady-state solution to theproblem, which has the interface at s = 8/11 ≈ 0.73. Evolution towardsthe steady-state (the dots) is apparent. In Figure 2.16, we plot the interfacelocation as a function of time. The stepwise behaviour of the location is dueto the definition that we choose. For simplicity, we take the interface locationto be the grid point to the left of the first positive value of temperature. Soeach jump is by precisely one grid point. We have also plotted the interfacelocation for the associated Neumann freezing problem. While our problemis on a finite domain, we expect the influence of the boundary condition atz = D to be small for small times, and thus the interface location to behavelike that of the Neumann solution for small times. Indeed, Figure 2.16 showsgood agreement between our computed s(t) and the Neumann solution s(t) =β√t, for small t.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

z

tem

pera

ture

T

Temperature profiles with increasing time

temperature evolving towards steady−state (dots)

Figure 2.15: Evolution of numerical results using the enthalpy method.

In Figure 2.17, we show an important feature of enthalpy method so-lutions to Stefan problems, namely the stepwise temperature history at apoint. This an non-physical effect introduced by the rapid adjustments andsubsequent relaxations of the temperature each time the interface advancesby one grid point. This is explained in more detail in [3] (Chapter 4).

Finally, in Figure 2.18, we plot the interface location versus time fortwo travelling wave solutions to the Stefan problem. For each of the twowave speeds c = 1, 3, we plot solutions obtained using N = 80, 160 gridpoints, with num = 400, 1600 time-steps, respectively. Convergence to thecorrect travelling wave solution is suggested by the diagram, but a numericalconvergence study is necessary to demonstrate this convincingly.

In Tables 2.2-2.3, we plot the computed errors between exact travellingwave solutions with speeds c = 1, 3, and the enthalpy method solution, anddenote the errors in temperature and interface location by ‖ET‖1 and ‖Es‖1

respectively. In each case, we use explicit time-stepping, with µ = k/h2

fixed, as we use a second order scheme. The number of grid points is N , andnum is the number of time steps. We quantify the errors by calculatingnumerically the norms of the errors made in the temperature T , and theinterface location s. Specifically, we compute the time-averaged quantities


0 0.5 1 1.5 2 2.5 3 3.5

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time t

inte

rfac

e lo

catio

n s(

t)

Interface position calculated using enthalpy method

interface locationβ t0.5

Figure 2.16: Evolution of the interface using the enthalpy method, togetherwith associated Neumann result.

Table 2.2: Errors for the enthalpy method, Forward Euler time-stepping,with c = 1.

N num ‖ET‖1 factor ‖Es‖1 factor

20 25 0.8604 6.3693E-840 100 0.4259 2.02 2.7862E-8 2.2980 400 0.2057 2.07 1.3431E-8 2.07160 1600 0.1011 2.04 6.3753E-9 2.11

defined by

‖ET‖1 =1

num

1

N

num∑

n=1

N∑

j=1

|T (zj, tn) − Texact(zj , tn)| ,

‖Es‖1 =1

num

num∑

n=1

|s(tn) − ctn| .

We notice that, despite a second order spatial discretization, the we do nothave second order accuracy. The errors decrease by a factor of about 2each time the grid spacing is halved, rather than a factor of 4. This is to be


0 0.5 1 1.5 2 2.5 3 3.5

x 107

−1

−0.5

0

0.5

1

1.5

2

2.5

3

time t

tem

pera

ture

T(0

.5,t)

Temperature history at a point, by the enthalpy method

Figure 2.17: Temperature history at the point z = 0.5.

Table 2.3: Errors for the enthalpy method, Forward Euler time-stepping,with c = 3.


20 25 1.4917 4.2443E-840 100 0.7783 1.92 2.0526E-8 2.0680 400 0.4283 1.82 1.0002E-8 2.05160 1600 0.2264 1.89 4.8610E-9 2.06

expected; due to the stepwise temperature histories, and the fact that we havenot dealt with the interface explicitly, lower order errors are introduced nearthe interface. Further details are available in [3]. Methods are available forimproving the accuracy of the computed interface location [65], based uponthe known phase-change temperature, and extrapolation methods. Secondorder accuracy in the interface location may be achieved, but not in thetemperature.

In our final convergence study in Table 2.4, we show the errors calculatedusing the enthalpy method for the Neumann freezing problem, where wecompute on z ∈ (0, 1), giving the exact Neumann solution as the boundarycondition at z = 1. The factor by which the temperature error decreases isnow around 2.6.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10−7

0

1

2

3

4

5

6x 10

−7

time t

inte

rfac

e lo

catio

n s(

t)

Interface location versus time for exact and numerical travelling waves

N=80, num=400N=160, num=1600exact solution

c=1

c=3

Figure 2.18: Computed interface location for travelling wave conditions, withc = 1 and c = 3.

Table 2.4: Errors for the enthalpy method, Forward Euler time-stepping, fora Neumann problem.


20 200 0.0595 0.014540 800 0.0229 2.60 0.0068 2.1380 3200 0.0087 2.63 0.0033 2.06160 12800 0.0034 2.56 0.0016 2.06

2.3 The Porous Medium Equation

In this section, we discuss some results and features of the Porous MediumEquation. This is a nonlinear diffusion equation which arises in fluid flow andother applications, which may result in the appearance of a moving interface.

2.3.1 Examples and applications

Consider the flow of an isothermal, ideal gas through a porous medium. Theflow obeys Darcy’s Law (see, for example [8]), which says that the volumetricflow rate of the gas through the medium is proportional to the pressure


gradient in the gas. Specifically, the volumetric flow rate, known as theDarcy velocity, is given by

u = −κµ∇p, (2.72)

where p is the gas pressure, κ is the permeability of the porous medium, andµ is the viscosity of the gas. For illustration in this section, we shall only beconcerned with one-dimensional problems, so that

u = −κµpz, (2.73)

where z is the space variable. Now, conservation of mass requires that

(φρ)t + (ρu)z = 0, (2.74)

where φ is the porosity of the medium, which we will assume to be constant,and ρ is the density of the gas. Now, the ideal gas law relates pressure anddensity such that

p =R

MρT, (2.75)

where R and M are the universal gas constant and the molar mass of thegas respectively, and T is the temperature. Thus, for isothermal gas flowthrough a porous medium, we find that

ρt = c(ρρz)z, (2.76)

for a constant c. This is a nonlinear diffusion equation with a variable diffu-sion coefficient ρ. Notice that it will be of parabolic type, provided that ρ > 0.Next, we consider problems where the diffusion coefficient may vanish.

Consider the motion of a thin liquid droplet on a solid substrate, shown inFigure 2.19. In the absence of any air flow driving the motion, and assumingthat surface tension effects are negligible, then the droplet will spread underthe effect of gravity alone. The free surface of the droplet is denoted y =h(x, t), and this surface meets the substrate at x = L(t) on the right handside. The thin film approximation of the Navier-Stokes equations gives anequation for the height h of the droplet (see [53]):

ht − (h3hx)x = 0. (2.77)


x

y

y = h(x, t)

x = L(t)

Figure 2.19: A thin droplet spreading under gravity over a solid substrate.

Here, finding the position of the wetting front L(t) will be part of the problem.We will see similar equations for conservation of liquid mass flowing throughporous media in later chapters. Notice here that the equation ceases to be ofparabolic type at the point x = L(t), where h = 0. This type of degeneratediffusion problem can lead to solutions with singular gradients.

Both (2.76) and (2.77) are examples of the Porous Medium Equation,the general form for which is

ut = (unux)x, (2.78)

where n > 0. In this thesis, we are particularly interested in problemswith n = 3. The spreading droplet example above suggests that equationsof this type admit solutions with compact support. Next, we will describesome well known analytical solutions.

2.3.2 Analytical solutions

Here, we will mostly concentrate on the Porous Medium Equation (2.78) withn = 3. First of all, let us consider the steady-state problem. That is,

(u3ux)x = 0. (2.79)

Writing this as1

4(u4)xx = 0,

we readily obtain a general solution

u(x) = A(L− x)1/4,


for constants A and L. Further, the solution

u(x) =

A(L− x)1/4 x < L,

0 x ≥ L,(2.80)

satisfies (2.79) in a classical sense everywhere except the free interface x =L. A sketch is shown in Figure 2.20. The derivative ux is undefined atx = L, and we regard this solution as a weak, or generalized solution. Weaksolutions satisfy an integral form of the equation, in the same way as weaksolutions to the Stefan problem. We will define a weak solution of the time-dependent problem shortly. For now, suppose we want to satisfy a boundarycondition u(0) = u0, and that we must have a total “mass” W in the system.That is, ∫ L

0

u(x) dx = W.

Then the constants A and L are easily found:

L =5

4

W

u0, A =

(4

5

u50

W

)1/4

.

x

y

u(x) = A(L− x)1/4

L

Figure 2.20: A solution of the steady-state Porous Medium Equation.

Now let us consider the time-dependent problem,

ut = (u3ux)x, (2.81)

and the sketch of a solution profile in Figure 2.21. Again, we seek solutionswith compact support. Any such solutions with discontinuous derivativesat the moving interface must be regarded as weak solutions. Such weak


x

y

u(x, t)

x = L(t)

Figure 2.21: A solution of the time-dependent Porous Medium Equation.

solutions are discussed in detail by Elliott and Ockendon [23], and we willnot reproduce their work here. We simply state, following the summarygiven in [43], that a weak solution of the problem (2.81) is a continuous,non-negative function u(x, t) for all t ≥ 0 and for all x, such that

∫ T

0

∫ ∞

−∞

u3

3ψxx + uψt dx dt+

∫ ∞

−∞ψ(x, 0)u(x, 0) dx = 0, (2.82)

for all test functions ψ which vanish at infinity and t = T , and which havecontinuous first derivatives. We shall not discuss the theory of weak solutionsfurther. Rather, we now seek to construct solutions.

Assuming that we can find a solution with compact support, such as thatshown in Figure 2.21, what structure will it have near the moving inter-face x = L(t)? One way to answer this is to suppose that the solution toequation (2.81) takes the form

u ∼ f(t)(x− L)r, r > 0, as x→ L(t)−. (2.83)

Substituting, we have

f ′.(x− L)r + f.r(x− L)r−1 L(t) ∼ f 4.r(4r − 1)(x− L)4r−2.

Then, for a nonzero, finite speed L(t), we require the balance

r − 1 = 4r − 2, ⇒ r =1

3.

Alternatively, a more formal approach considers conservation of “mass” atthe interface (see [54], for example), which gives

L(t) = − limx→L(t)−

(u3)x. (2.84)


Substituting the form (2.83) again gives r = 13

for a nonzero, finite speed. Sothe solution needs to have this certain singularity structure in order for theinterface to move.

Barenblatt-Pattle “Spreading blob” solutions

The Barenblatt-Pattle solution of the Porous Medium Equation is a similaritysolution of the form

u(x, t) = tαF (η), η =x

tβ.

A number of authors have investigated the behaviour of such solutions; here,we follow the introduction given by Lacey et al [43]. The values of α and β arefound by the conditions that the solution be self-similar, and that the totalmass contained within a “blob” is conserved. For the general equation (2.78),we find

α =−1

2 + n, β =

1

2 + n.

For the case n = 3, the solution is

u(x, t) =

(310

)1/3t−1/5

(a2 − x2

t2/5

)1/3

, |x| < at1/5,

0, |x| ≥ at1/5,(2.85)

where a is a constant depending on the total mass in the system. In Fig-ure 2.22, we show some solutions with a = 1, for various times, and thespreading blob shape is clear.

Travelling wave solutions

As for the Stefan problem, we can also construct a travelling wave solution.For a travelling wave speed c, we seek solutions to (2.81) of the form

u(x, t) = f(ξ), ξ = x− ct.

The problem reduces to the ODE

−cf ′ = (f 3f ′)′,

which has a solutionf(ξ) = (−3cξ)1/3.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Barenblatt−Pattle "spreading blob" solutions of the PME, with a=1

x

u

t=0.1t=5t=15

Figure 2.22: Barenblatt-Pattle solutions of the Porous Medium Equation.

Thus, our travelling wave solution is

u(x, t) =

(−3c(x− ct))1/3 , x < ct,0, x ≥ ct.

(2.86)

Now, as before, we continue by describing numerical methods for the solu-tion of Porous Medium Equation interface problems, and will return to ouranalytical solutions to evaluate the performance of these methods.

2.3.3 Numerical methods

The moving interface problems arising from the Porous Medium Equation re-quire special care when developing numerical schemes for their solution. Thesingular gradients at the interface may introduce problems, while conserva-tive schemes are key to finding a solution with the correct interface velocity.Consider the equation (2.81). The gradient ux becomes singular at the inter-face where u = 0. To resolve the sharp interface, adaptive gridding may beemployed to add resolution near the interface. However, we aim to developa capturing method which, like the enthalpy method for the Stefan problem,does not require any explicit handling of the interface. One approach, whichhas appeared for similar problems in the fuel cell literature (see [50, 51],for example), is a computational regularisation. This amounts to replacingequation (2.81) with

ut =((u3 + η)ux

)x, (2.87)


where η > 0 is a computational parameter. The effect of this is to removethe degeneracy and singularity from the problem at u = 0, as the diffusioncoefficient remains positive. Thus, the problem remains strictly parabolic,and the sharp interface is smeared out. As an alternative, we consider themethod described by Evje and Karlsen [25].

Consider the problem

ut = (u3ux)x , 0 < x < 1, t > 0,u(x, 0) = u0(x),−u3ux|(0,t) = QL,−u3ux|(1,t) = QR,

(2.88)

where QL,R are fluxes at the left and right hand end points. Now considera finite difference discretization of the problem, with time-step k, h = 1/Nand Un

j is the approximation for u((j − 1)h, (n − 1)k), for j = 1, .., N + 1.In [25], it is stated that a naive discretization of the kind

Un+1j = Un

j + kh

(Un

j +Unj+1

2

)3 (Unj+1

−Unj

h

)

−(Un

j−1+Un

j

2

)3 (Unj −Un

j−1

h

), j = 2, .., N,

(2.89)

will not necessarily result in a conservative scheme. The alternative ideapresented in [25] is to rewrite the problem as

ut = 14(u4)xx, 0 < x < 1, t > 0,u(x, 0) = u0(x),

−14(u4)x|(0,t) = QL,

−14(u4)x|(1,t) = QR,

(2.90)

and then discretize, using standard centered differencing for (u4)xx. We nowshow how a conservative scheme results, using a ghost point method for theboundary fluxes. For illustration, we use Forward Euler time-stepping, butthe same argument will follow for implicit schemes.

Step in time using the following scheme:

Un+1j = Un

j +µ

4

((Un

j−1)4 − 2(Un

j )4 + (Unj+1)

4), j = 2, .., N, (2.91)

where µ = k/h2. Now derive discrete boundary conditions, using a ghostpoint method:


Un+11 = Un

1 + µ2

((Un2 )4 − (Un

1 )4) + 2 khQL

Un+1N+1 = Un

N+1 + µ2

((Un

N)4 − (UnN+1)

4)− 2 k

hQR.

(2.92)

The scheme must conserve “mass” at each time step. That is, we must satisfy

∫ 1

0

u(x, nk) dx =

∫ 1

0

u(x, (n+ 1)k) dx,

in a discrete sense. Suppose we require that our scheme conserves mass usingtrapezoidal approximation for the integral. Then we have

massn+1 = h2Un+1

1 + h(∑N

j=2Un+1j

)+ h

2Un+1N+1,

= h2

Un

1 + 2 khQL + µ

2((Un

2 )4 − (Un1 )4)

+h∑N

j=2

Unj + µ

4

((Un

j−1)4 − 2(Un

j )4 + (Unj+1)

4)

+h2

µ2

((Un

N)4 − (UnN+1)

4)

+ UnN+1 − 2 k

hQR

,

= h2Un

1 + h(∑N

j=2Unj

)+ h

2UnN+1 + k(QL −QR)

+hµ4

(Un

2 )4 − (Un1 )4 +

∑Nj=2

((Un

j−1)4 − 2(Un

j )4 + (Unj+1)

4)

+(UnN)4 − (Un

N+1)4.

That is,

massn+1 =h

2Un

1 + h

(N∑

j=2

Unj

)+h

2UnN+1 + k(QL −QR) + S, (2.93)

where

S = hµ4 (Un

2 )4 − (Un1 )4

+∑N

j=2

((Un

j−1)4 − 2(Un

j )4 + (Unj+1)

4)

+ (UnN)4 − (Un

N+1)4.

(2.94)


We recognize the sum in (2.94) as a telescoping sum, and find that S = 0.Hence, equation (2.93) gives

massn+1 = massn + k(QL −QR), (2.95)

which says that mass is conserved.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Profiles obtained using conservative and nonconservative schemes for ut=(u3u

x)x

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

x

u

conservativenonconservativeexact

Figure 2.23: Evolution of profiles given by conservative and non-conservativeschemes applied to ut = (u3ux)x.

In Figure 2.23, we show a succession of profiles obtained numerically usinga conservative scheme, and a non-conservative scheme, applied to the sameproblem. While both give solutions with compact support which “look rea-sonable”, the solution obtained using the non-conservative scheme is clearlywrong. For problems with degenerate diffusion-type terms in later chapters,we shall use Evje’s spatial discretiaztion to help ensure conservative schemes.


In Tables 2.5-2.6, we show numerical convergence studies for solutions to (2.81)on −0.5 < x < 0.5, with travelling wave solutions from (2.86) as initial and


boundary conditions. We use Evje’s discretization (2.91). Specifically, wecompute the time-averaged errors in computed values of u, and the interfacelocation L, defined by

‖Eu‖1 =1

num

1

N

num∑

n=1

N∑

j=1

|u(zj, tn) − uexact(zj, tn)| ,

‖EL‖1 =1

num

num∑

n=1

|L(tn) − ctn| .

As with the solutions of the Stefan problem, we have used a second ordercapturing scheme to approximate solutions of a moving interface problem,in which we do not treat the interface explicitly at all. Again, we do notachieve second order accuracy, but a factor of around 2.4-2.6 decrease foreach halving of the grid space is clear.

Table 2.5: Errors for Evje’s method, Forward Euler time-stepping, for thePorous Medium Equation on −0.5 < x < 0.5, with µ = 0.1 and c = 3

N num ‖Eu‖1 factor ‖EL‖1 factor

20 50 1.348E-2 2.122E-240 200 5.217E-3 2.58 1.165E-2 1.8280 800 2.034E-3 2.56 5.689E-3 2.05160 3200 8.297E-4 2.45 3.087E-3 1.84

Table 2.6: Errors for Evje’s method, Forward Euler time-stepping, for thePorous Medium Equation on −0.5 < x < 0.5, with µ = 0.5 and c = 0.5

N num ‖Eu‖1 factor ‖EL‖1 factor

20 100 7.058E-3 2.071E-240 400 2.855E-3 2.47 1.151E-2 1.8080 1600 1.140E-3 2.50 5.723-3 2.01160 6400 4.652E-4 2.45 3.040E-3 1.88

60

Chapter 3

Nonisothermal, steady-statephase change in porous media

In this chapter, we describe the steady-state model problem of phase changein a sand pack, as presented by Udell [64]. His experimental results are ob-tained by partially saturating a sand-filled tube with water, and then heatingfrom the top, while cooling from the bottom, and measuring temperaturesalong the height of the pack. At steady-state, the system supports fluid in

heat

cool

vapour

two−phase

liquid

z

z=D

z=L

z=0

Figure 3.1: Udell’s experiment.

the pore space in one of three configurations:

1. Only vapour throughout the entire pack.

2. A vapour-only zone above a two-phase zone in which both liquid andvapour exist.

3. A vapour-only zone above a two-phase zone, with a liquid-only zoneunderneath.

Chapter 3. Nonisothermal, steady-state phase change in porous media 61

A three-zone system is shown in Figure 3.1. Udell [64] presents experimen-tal results for the three-zone system, where the two-phase zone is identifiedas the almost isothermal region between two single-phase regions with lin-ear temperature profiles. He then presents an analysis of the two-phasezone only, under the assumption that it is isothermal. We can use Udell’ssteady-state two-phase zone model as a starting point, which can then beextended to the free interface problem for a two or three zone system. Adisjoint-domain computational method for locating the interfaces in a one-dimensional, steady-state, three-zone (vapour/two-phase/liquid) system isdescribed by Bridge et al [15]. It is worth noting that their model relaxesthe popular assumptions of isothermal two-phase zones and phase changeonly at the boundaries, and we wish to keep temperature and condensationrate effects in this work.

Motivated by the fuel cell setting, where regions of liquid water only donot occur [12, 21, 56, 58, 70], we will concentrate here on a two-zone systemwhich consists of a two-phase zone and a vapour-only zone. We presenta computational method, based on that in [15], to compute the interfacelocation in such a steady two-zone system. The numerical results obtainedfrom this method will be used as benchmarks with which we will compare theresults from the unsteady computations to be described in the next chapter.

Also in this chapter, we consider the method of Residual Velocities, pro-posed by Donaldson [22] for the steady-state Stefan problem and describedbriefly here in Chapter 2, and show how this method may be used to computesteady-state solutions to our model phase-change problem.

3.1 Mathematical formulation of the model

problem

In Figure 3.2, we show a schematic of the basic setup, as used by Udell [64].The porous layer is initially saturated with a certain amount of liquid water,then heated from above and cooled from below. A steady-state is realised,with two distinct zones appearing. In the two-phase zone, z ∈ (0, L), liquidand vapour coexist in the pore space. In this region, the liquid is drivenupwards by capillary pressure, while the vapour is driven downwards by avapour pressure gradient. In the vapour-only region, z ∈ (L,D), the watervapour is stationary. A primary goal of the works by Udell [64] and Bridge


liquid vapour

vapour only

two−phase

heat

cool

z=D

z=L

z=0

Figure 3.2: A two-zone system for Udell’s experiment.

et al [15] is to locate the interface z = L. We note that the model presentedin [15] assumes a constant vapour density throughout the two-phase andvapour-only regions. Here, we allow for compressible vapour, in accordancewith the Ideal Gas Law. The method we use is then adapted from thatused in [15]. A list of physical constants and parameters used in our modelappears in Table 3.1.

In order to locate the interface, we consider the saturation s through theporous layer. The saturation is defined by

s =volume of pore space occupied by liquid water

volume of pore space. (3.1)

Therefore, a liquid-only region (which we do not consider here) has s ≡ 1, avapour-only region has s ≡ 0, and a two-phase region of vapour and liquidhas 0 < s < 1. In this model, we assume that s is continuous throughout theporous layer, and that at the interface z = L, we have s → 0+ as z → L−.Now, we formulate a model for the saturation s (the liquid volume fraction),and the temperature T , and their variations in z, the height up the porouslayer. We need to consider energy conservation and mass conservation ineach of the two zones. First, we consider the two-phase region 0 < z < L.


Table 3.1: Constants for Udell problem

Symbol Interpretation Typical value Units (SI)

φ porosity 0.38 -κ permeability 6.4 × 10−12 m2

ρℓ liquid density 103 kg/m3

cv specific heat of vapor 103 J/kg Kcℓ specific heat of liquid water 4.2 × 103 J/kg Kµv viscosity of water vapour 2.2 × 10−5 kg/msµℓ viscosity of liquid water 2.5 × 10−4 kg/ms

Kv thermal conductivity ofvapor saturated medium 1.0 W/mK

ρc mass averageddensity heat capacity product 105 J/K m3

hvap latent heat (water liquid-vapor) 2.5 × 106 J/kgδ capillary pressure scaling 1.7642 × 104 PaR universal gas constant 8.31 J/moleKM molar mass of water 18 × 10−3 kg/molea characteristic vapour pressure 0.19743 Pab characteristic temperature 0.03525 K−1

q heat flux ∼ 103 W/m2

Conservation of liquid mass gives

(ρlul)z = Γ. (3.2)

Here, ρ is density, u is the superficial Darcy velocity, and the subscript ldenotes liquid. The source term Γ is the rate of production of liquid water, sowe shall refer to this as the condensation rate. In the same way, conservationof vapour mass in the two-phase zone reads

(ρvuv)z = −Γ, (3.3)


where subscript v now denotes vapour. The term on the right hand side isthe rate at which vapour mass is produced. That is, −Γ is the evaporationrate. Now, conservation of energy, neglecting convective effects as in [15], isgiven by

0 =(KTz

)z+ hvapΓ, (3.4)

where K is the effective thermal conductivity of the liquid-vapour saturatedporous medium, and hvap is the specific heat of vaporization of water. Asin [15], we will neglect saturation effects on the thermal conductivity, andassume a value for this mass-averaged quantity.

The Darcy velocities ul, uv are the average volumetric flow rates of liquidand vapour, respectively, through the pore space. Darcy’s law gives relationsbetween the velocities and the pressure gradients in the two fluid phases asfollows:

ul = −κκrlµl

((pl)z + ρlg) , (3.5)

uv = −κκrvµv

(pz + ρvg) . (3.6)

Here, κ is the permeability of the porous medium, pl is the liquid pres-sure, p is the vapour pressure, and g is the acceleration due to gravity. Thequantities κrl and κrv are the relative permeabilities of liquid and vapourrespectively. These relative permeabilities account for the decrease in mo-bility of one phase due to the presence of another, and hence depend on thesaturation s. In particular, we require that κrl is unity when only liquid ispresent, and zero in pore space occupied by vapour only. That is, κrl shouldbe an increasing function of s. By similar reasoning, we require that κrv isa decreasing function of s. Here, we will use the cubic forms suggested byUdell [64], following the empirical results of Fatt and Klickoff [28]:

κrl = s3, (3.7)

κrl = (1 − s)3. (3.8)

We now seek relationships between between the pressures and the primaryvariables s and T . A major assumption of the model presented in [15] is thatthe vapour in the two-phase region is fully saturated. In keeping with thismodel, we will assume here that the vapour is fully saturated, and that the


temperature and saturation pressure, psat approximately obey the exponen-tial relation

psat(T ) = aebT , (3.9)

where the constants a and b are fitted to data from saturated steam tablesin [1], for example. Then, in the two-phase zone, we take p = psat(T ), asexplained by Baggio et al [7]. Now we present a constitutive law for theliquid pressure. At the pore scale, the interfacial tension between liquid andvapour phases gives rise to capillary effects. The capillary pressure, pc, isdefined as the difference between the vapour and liquid pressures,

pc = p− pl. (3.10)

As shown by Leverett [45], the capillary pressure is found to be a function ofthe saturation. The functional form for the capillary pressure, pc = pc(s) isknown as the Leverett function. Udell [64] correlates this Leverett functionto write the capillary pressure as

pc(s) = δ J(s), (3.11)

where J(s) is given by

J(s) = 1.417(1 − s) − 2.120(1 − s)2 + 1.263(1 − s)3, (3.12)

and

δ = σ

(φ

κ

) 1

2

,

where σ is the vapour-liquid interfacial tension, and φ is the porosity of theporous medium. Also, we should note that the function J(s) given in (3.11) isan empirical relationship which will depend on the particular porous mediumbeing used. The correlation given in [64] is for a particular type of sand.However, in the absence of another model, and to allow comparison with ourresults, we will continue to use this model.

Finally, in the two-phase zone (and indeed throughout the entire system),we assume that the water vapour obeys the ideal gas law, namely

p =R

MρvT, (3.13)

where R and M are the universal gas constant and the molar mass of waterrespectively. The work presented by Bridge et al [15] assumes a constant


vapour density, but we wish to include compressibility due to thermal effectshere. In summary, the two-phase zone conservation equations (3.2)-(3.4),together with all the constitutive relations and empirical laws, form a secondorder system in three unknowns s, T and Γ. Furthermore, we can eliminatethe condensation rate Γ by summing (3.2) and (3.3), arriving at the system(

ρl

µls3(ddz

(psat(T ) − δJ(s)) + ρlg)

+ MRµv

psat(T )T

(1 − s)3(ddz

(psat(T )) + MgR

psat(T )T

) )z

= 0,

(3.14)(KdT

dz+ κhvap

M

Rµv

psat(T )

T(1 − s)3

(d

dz(psat(T )) +

Mg

R

psat(T )

T

))

z

= 0.

(3.15)Equations (3.14) and (3.15), form a coupled, second order, nonlinear systemfor the two-phase zone variables s and T , in the region 0 < z < L. Thesystem clearly has a singularity and degeneracy at s = 0. In the case ofconstant vapour density, Bridge [14] shows that s = O(L− z)1/4 as z → L−,and as such, the degenerate diffusion type term requires careful treatmentwhen constructing a numerical solution.

The vapour-only zone L < z < D is more simple. The temperature isharmonic, and the saturation is zero everywhere in this zone. Conservationof mass reads

(ρvuv)z = 0, (3.16)

while conservation of energy reads

0 = Tzz, (3.17)

since there is no phase change in the vapour-only region. Again, we assumethat the vapour is ideal, and the system (3.16)-(3.17) can be cast as a systemfor the temperature T and vapour density ρv in the region L < z < D.For this one-dimensional problem, exact solutions are available in terms ofthe interface location L, the fixed boundary temperature T (D) = T1, andthe temperature and pressure at z = L+, which we will denote T+ and p+

respectively. The temperature in the vapour region is given by

T (z) = T+ +T1 − T+

D − L(z − L),

orT (z) = T+ +

q

Kv

(z − L), (3.18)


where Kv is the thermal conductivity of the vapour-saturated porous medium,and q is the constant heat flux through the porous layer. Notice that, if thevapour in this zone is stationary, then the vapour pressure in the vapour-onlyregion is the solution of a separable first order equation, and is given by

p(z) = p+

(T (z)

T+

)−MgKv/Rq

. (3.19)

The vapour density in this region may be written

ρv(z) = ρ+

(T (z)

T+

)−(1+ MgKvRq

)

. (3.20)

These exact solutions will be useful when implementing both the disjointdomain method described in [15], and the Residual Velocity Method proposedby Donaldson [22], where we can iteratively solve problems parameterized byheat flux.

Now we examine the boundary and interface conditions required to closethe model problem. We have elliptic problems in two variables in the tworegions, so we shall specify two conditions on the two physical boundaries z =0 and z = D, and four conditions on the interface z = L. Since the interfacelocation L is also an unknown in the problem, we require a fifth condition atthe interface.

At the boundary z = D, we impose a temperature T1, say. Also, sincewe consider a closed porous pack, there can be no mass flux across z = D.Hence, at z = D, we have

T = T1, (3.21)

uv = 0. (3.22)

Now, at the free interface, we require five conditions. First of all, the sat-uration s is zero. Temperature and vapour pressure should be continuous.Finally, mass and energy should both be conserved across the interface, sothat the heat flux is continuous, and there is no mass flux across the interface.


In summary, at z = L, we have the following five conditions:

s = 0, (3.23)

[T ]+− = 0, (3.24)

[p]+− = 0, (3.25)

(ρlul + ρvuv)− = (ρvuv)

+ , (3.26)(KTz − hvapρvuv

)−=(KTz

)+

. (3.27)

We note here that condition (3.27) corresponds to a singular evaporation rateat the interface. At the boundary z = 0, we have an imposed temperatureand zero mass flux, which give

T = T0, (3.28)

ρlul + ρvuv = 0. (3.29)

We have specified five conditions at the interface, but we see that a uniquesolution will not be available. Imposing the boundary condition (3.29) leavesthe interface condition (3.26) redundant, and a further condition is required.In order to determine the vapour pressure uniquely throughout the porouslayer, we now impose a global integral constraint on the system. Experi-mentally, the total water mass in the system can be controlled, and will beknown. Suppose that the fixed water mass per cross sectional area of thesaturated porous pack is W . Then we have the integral constraint

∫ L

0

(sρl + (1 − s)ρv) dz +

∫ D

L

ρv dz = W, (3.30)

which closes the system. It is clear that the three control parameters forthe experiment are the temperatures T0, T1 and the water mass W . In thenext section, we will describe a numerical method which, given these threeparameters, will calculate the location of the free interface z = L.

3.2 An iterative disjoint-domain method

The system we are trying to solve is summarized in Figure 3.3. In [15], analgorithm is described for solution of the steady-state problem with a liquidzone, a two-phase zone, and a vapour zone, where there are two interfaces


to find. Here, there is just one interface. There is an additional nonlinearityin the model here, as we have allowed for compressible vapour. Now we willdescribe an adaptation of the existing method to solve our problem for theinterface location L.

z = D

z = 0

vapour, (p, T )

mass equation (3.14)

energy equation (3.15)

interface conditions(3.23), (3.24), (3.25),(3.27)

z = L

two-phase, (s, T )

T = T1, ρlul + ρvuv = 0

T = T0, ρlul + ρvuv = 0

mass equation (3.16)

energy equation (3.17)

With integral constraint (3.30).

Figure 3.3: System for steady-state solution of the free interface problem.

Our solution method is parameterized by a heat flux q and the saturationat z = 0, which we denote s0. Now, if the heat flux across the boundary z = 0is q, then we have

KTz − hvapρvuv = q, at z = 0.

Notice that this then allows us to integrate the energy equation (3.15) once,leaving

KTz − hvapρvuv = q, for 0 < z < L. (3.31)

Also, taking the zero mass flux condition at z = 0 allows us to integrate the


mass equation (3.14) once, leaving

ρlul + ρvuv = 0, for 0 < z < L. (3.32)

Thus, given q, we reduce the problem in the two-phase zone to a coupledsystem of ordinary differential equations,

ρl

µls3(ddz

(psat(T ) − δJ(s)) + ρlg)

+ MRµv

psat(T )T

(1 − s)3(ddz

(psat(T )) + MgR

psat(T )T

)= 0,

(3.33)

KdT

dz+ κhvap

M

Rµv

psat(T )

T(1 − s)3

(d

dz(psat(T )) +

Mg

R

psat(T )

T

)= q,

(3.34)which will be solved as an initial value problem for s and T , with initialvalues s0, T0. This initial value problem in z is solved numerically until s = 0.This requires some care, since the coefficient s3 in the first term of (3.33)makes the problem singular at s = 0. One approach is to reformulate andsolve for z(s) rather than s(z), which is the method we have used here.Another approach is to make the change of variables w = s4, which leaves aregular problem for w. This idea has already been discussed with respect tonumerical methods for degenerate diffusion problems, and will feature againin the next chapter.

Once we have solved until s = 0, then we stop, and we have found theinterface location L. Then it remains to solve the problem for pressure andtemperature in the vapour region, subject to the remaining boundary, in-terface and global conditions. Given continuity of temperature (3.24) andpressure (3.25), we find the values of temperature and pressure just abovethe interface, namely T+ and p+. Also applying the heat balance (3.27) at theinterface, and no mass flux at z = D, we find that the problem (3.16)-(3.17)has the analytical solution (3.18)-(3.19).

So, given q and s0, it is relatively straightforward to solve the system ofequations and find the interface L. However, the condition T = T1 at z = 0,and the global integral constraint on the mass W (3.30) have yet to besatisfied. The idea is to iterate on q and s0 until these two conditions aresatisfied. Suppose that, for given q and s0, we have computed a solutionusing the algorithm described here, which has T = T at z = D, and whichhas a total water mass (per unit area) of W . Then, defining

G(q, s0) =

(T − T1

W −W

), (3.35)


we seek zeros of the function G. Following [15], we compute the zeros usinga quasi-Newton method which uses centered differences for the numericalderivatives.

3.2.1 Numerical results and discussion

Our Newton iteration is seen to be quite sensitive to initial guesses. That is,we require a good initial guess in order for the iterations to converge. Goodinitial guesses in many of our computations have been generated by a methodof continuation, similar to that described by Zwillinger in [75] (Chapter 168).Suppose we have base values of the control parameters T0, T1 and W , forwhich we have a numerical solution with a two-zone system, and “target”values of these parameters, for which we wish to solve a new problem. Wetake the values of q and s0 corresponding to the base values as initial guessesfor a problem with slightly adjusted values of the control parameters. Thenumerical solution of the resultant problem generates new values of q and s0,which are then used as initial guesses for the next slightly adjusted problem.This process is repeated as we gradually adjust the control parameters totheir target values. In practice, we have adjusted the control parametersone at a time. That is, we have continued in the parameters T0, T1 and Windependently.

In Figure 3.4, we plot the numerical solutions obtained using the iterativemethod, with T0 = 320, T1 = 450 and W = 30. Our computations have usedthe parameter values given in Table 3.1, together with the height of the sand-pack used by Udell [64], D = 0.254m. Also, we have neglected gravity effects,so that g = 0, and we will continue to do so throughout the remainder ofthis work. Notice that, by (3.19), this gives a constant pressure throughoutthe vapour-only region. The interface L ≈ 0.12 is clear, as is the singularityin the saturation gradient as we approach the interface z → L−. We makenote of the vivid variation in the vapour density ρv, which is a feature notincluded in the previous work [15]. Also, the temperature in the two-phaseregion is not constant. Two-phase regions in porous media are often modelledas being isothermal, but we are particularly interested in capturing methodsfor nonisothermal phase change problems. To underline the effect of notmaking the isothermal assumption, in Figure 3.5, we plot a close-up of thetemperature profile, showing a variation of about 7 K over the two-phaseregion. The structure of the temperature profile in this region is the same asthat described in [15].


0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

satu

ratio

n, s

Steady−state, compressible Udell problem

0 0.05 0.1 0.15 0.2 0.25

0.1

0.105

0.11

0.115

0.12

0.125

0.13

vapo

ur d

ensi

ty, ρ

v

T1=450, T0=320, q=928.4, s0=0.78, L=0.12

0 0.05 0.1 0.15 0.2 0.25320

340

360

380

400

420

440

z

tem

pera

ture

, T

0 0.05 0.1 0.15 0.2 0.25

1.6

1.7

1.8

1.9

2

x 104

z

vapo

ur p

ress

ure,

p

Figure 3.4: Solution profiles for free boundary problem using the iterativedisjoint-domain method.

In computing the solutions and interface location, we have eliminated thecondensation rate Γ. The structure of this function is of interest. Using (3.4),we calculate the condensation rate for T0 = 320, T1 = 450 and W = 30, andplot Γ in the two-phase zone in Figure 3.6. The condensation rate is zeroin the vapour-only region. In the two-phase region, far from the interface,Γ is positive, and increasing towards the cold lower boundary, as we wouldexpect. As we approach the interface L, we know that the condensation ratebecomes singular and negative, signifying an evaporation front. The liquidwater that is driven upwards towards the interface evaporates very quickly,and the resulting vapour is then driven downwards. As in [15], the phasechange is concentrated in a layer near the cold boundary and at a frontat the interface between the two-phase and vapour zones, but is nonzerothroughout the two-phase region.


0.02 0.04 0.06 0.08 0.1 0.12 0.14320

325

330

335

340

345

z

tem

pera

ture

, T

Structure of temperature profile

Figure 3.5: Close-up of temperature profile for free boundary problem usingthe iterative disjoint-domain method.

In Table 3.2, we show further numerical results for various values of thecontrol parameters T0, T1, and W . In the first three rows, we have T0 = 340and W = 20, while we vary the upper boundary temperature T1. Decreas-ing T1 corresponds to an decrease in the magnitude of heat flux throughthe porous layer. This results in an increase in the length of the two-phaseregion, as described in [14].

Table 3.2: The effect of varying boundary temperature.

T0 T1 W q s0 L

340 600 20 1595 0.51 0.092340 550 20 1356 0.41 0.100340 400 20 542 0.21 0.145375 670 36 2520 0.61 0.140375 550 36 1858 0.42 0.160375 500 36 1541 0.35 0.174

In Table 3.3, we show results of another numerical experiment. This time,


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

0

0.01

0.02

0.03

0.04

0.05

z

cond

ensa

tion

rate

Γ

Condensation rate in the two−phase zone

Figure 3.6: Condensation rate in the two-phase zone.

we fix the boundary temperatures, and vary the total water mass W . As Wincreases, the boundary saturation s0 increases, and so does the length L ofthe two-phase zone, as expected.

Table 3.3: The effect of increasing mass.

T0 T1 W q s0 L

320 450 15 790 0.29 0.093320 450 20 848 0.42 0.105320 450 25 893 0.61 0.113320 450 30 928 0.78 0.122

Finally, we note that the method may fail, if at the stage of comput-ing solutions to the two-phase zone initial value problem (3.33)-(3.34), thecomputed interface location L is greater than the height of the layer, D.This signifies that a two-phase zone is supported by the heat flux, withoutan accompanying vapour-only region. Indeed, given a heat flux and initialsaturation, (3.33)-(3.34) may be used to easily solve for the saturation andtemperature in a single-zone, two-phase system. Any numerical capturingmethod we develop should general enough so that it can deal with bothsingle and two-zone systems.


3.3 The method of residual velocities

In this section, we extend the method of residual velocities described in theprevious chapter to our steady-state phase change problem in porous media.Consider again the problem shown in Figure 3.3. We follow the same idea asfor the Stefan problem, by giving an initial interface position L, and solvingthe problem subject to all but one of the interface conditions. The computedresidual in this interface condition will be used as the interface “velocity”,and we step in “time” to evolve the interface to the correct steady-stateposition. There are four interface conditions to satisfy. For illustration here,we describe the method, using the residual in interface condition (3.24) forthe interface velocity.

For a given L, we solve the boundary value problems on either side ofthe interface by using an iterative method which will ensure that the globalconstraint (3.30) is satisfied. The method we describe below is again basedon one-dimensional integration which allows us to solve a series of initialvalue problems.

First, make a guess for the heat flux q and the temperature just below theinterface T−. Then s and T in the two-phase region come from the solution ofthe initial value problem given by the ordinary differential equations (3.33)-(3.34), together with the initial conditions T (L) = T−, s(L) = 0, and solvingon 0 < z < L.

Using the continuity of pressure (3.25), we have p+ = psat(T−), and then

the solution of the vapour-only problem is given by

T (z) = T1 +q

K(z −D), p = p+, 0 < z < D, (3.36)

so thatT+ = T1 +

q

K(L−D).

Bearing in mind that we are using (3.24) for the interface velocity, the twoconditions left to satisfy are T (0) = Tbot, and the mass constraint (3.30). Sowe iterate on q and T− until these are satisfied, using a quasi-Newton methodagain.

Once all the other conditions are satisfied, we define the interface “veloc-ity” to be

L(t) = T− − T+, (3.37)

and step in time using a numerical integrator.



0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

"time" t

inte

rfac

e po

sitio

n L(

t)

Evolution to correct steady−state using method of residual velocities

Figure 3.7: Convergence to correct interface location, using method of resid-ual velocities, for T0 = 320, T1 = 450, W = 30.

In Figure 3.7, we plot the interface location as a function of “time”, asgiven by the numerical solution of equation (3.37), for the problem withT0 = 320, T1 = 450 and W = 30, for three different initial values of theinterface position. The interface location is seen to evolve and converge tothe correct value L = 0.12, for all three of the initial values. In Figure 3.8,we plot the computed L(t) for the problem with T0 = 375, T1 = 500 andW = 36, for which we have previously computed an interface at L = 0.174.

3.3.2 A model problem in higher dimensions

The extension of this method to higher dimensions will require solutions ofboundary value problems on irregular domains, and a consideration of theinterface geometry. As discussed in earlier chapters, front-tracking methods,even such as this one which solves elliptic problems in the disjoint domains,often require significant computational effort in remeshing and updating theinterface position. We will not attempt to apply this method to our phasechange problem in higher dimensions. Instead, we propose a related modelproblem which falls into a class of free interface problems which have beenstudied by Chen and Wetton [19]. They discuss a class of problems which


0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

"time" t

inte

rfac

e po

sitio

n L(

t)

Evolution to correct steady−state using method of residual velocities

Figure 3.8: Convergence to correct interface location, using method of resid-ual velocities, for T0 = 375, T1 = 500, W = 36.

have a free interface which lies between two regions, in each of which a vectorLaplacian type problem must be satisfied.

To start with, consider a generalization of our current model problemto two dimensions, as shown in Figure 3.9. The five conditions at the freeinterface are now

s = 0, (3.38)

[T ]+− = 0, (3.39)

[p]+− = 0, (3.40)

(ρlul + ρvuv)− .n = (ρvuv)

+ .n, (3.41)(K∇T − hvapρvuv

)−.n =

(K∇T

)+

.n. (3.42)

where n is the unit normal to the interface.Now we consider a reduction of this model problem, as we seek a prob-

lem of vector Laplacian type in each subdomain Ω1,2, and interface condi-tions which are linear in the dependent variables. This class of problems isamenable to the linear analysis presented by Donaldson [22] for scalar prob-lems, and extended to vector problems in [19]. In the following reduction ofthe physical problem given in Figure 3.9 and (3.38)-(3.42), we replace mostof the physical constants with unity, and remove the nonlinearity in the gov-erning equations by assumptions and simplifications which leave a related,


Ω+ (vapour)

Ω− (two − phase)

∇.(ρvuv) = 0

T = 0

∇.(ρlul + ρvuv) = 0

∇.(K∇T − hvapρvuv) = 0

interface

conditions

(3.38)-(3.42)

Figure 3.9: Steady-state system in higher dimensions.

but not physical, model. Also, the reduced model lacks the singularity anddegeneracy associated with the physical problem. However, it is with a viewto future work in extending the residual velocity method to our nonlinearboundary value problems that we present this loosely related model.

In the vapour region Ω1, we already have Laplace’s equation for the tem-perature. Furthermore, if we assume that the vapour density is almost con-stant, then we approximate the mass equation by

p = 0.

In the two-phase region Ω2, we seek a saturation equation. Taking a con-stant relative permeability, assuming a linear function for the Leverett func-tion J(s), and just keeping the highest order term in s, we replace massconservation with

s = 0.

Finally, in Ω2, we replace the energy equation with

T = 0,


which is equivalent to neglecting phase change effects in this two-phase zone.We make similar reductions to the interface conditions (3.38)-(3.42), suchthat they still loosely represent continuity of saturation, temperature, pres-sure, mass flux and heat flux. The conditions we propose are:

s = 0, (3.43)

[T ]+− = 0, (3.44)

p+ = T−, (3.45)

(∇T + ∇s)− .n = (∇p)+ .n, (3.46)

(α∇T −∇s)− .n = (∇T )+ .n. (3.47)

A similar model problem has been studied in [19], and we leave the exten-sion of this model to include the nonlinear, singular features of the physicalproblem as future work.

3.4 Benchmark solutions

The steady-state solutions to the one-dimensional Udell problem that wehave described in this chapter will be useful in validating computed solu-tions to the time-dependent problem that we will study in the next chapter.We shall henceforth refer to solutions generated using the disjoint-domainmethod here as benchmark solutions. We will check that solutions of time-dependent problems with the same values of T0, T1 and W , with no massflux across the physical boundaries, evolve to the correct steady-state. Theimplementation of the interface capturing method that we develop in thenext chapter benefits from a relatively straightforward extension from onedimension to two dimensions, and our benchmark solutions will also be usedto validate solutions to the two-dimensional problem, in cases where we givetwo-dimensional initial data, but one-dimensional boundary data.

80

Chapter 4

The M2 mixture method forthe transient Udell problem

In this chapter, we describe the time-dependent extension of the steady, one-dimensional phase-change problem presented in Chapter 3. We carefullyderive the conditions on the moving interface, and show that these will bedifficult to implement in disjoint-domain numerical solution methods whichinvolve front tracking. Clearly, front capturing methods appeal for this typeof problem. We shall describe a reformulation of the problem over a fixeddomain, in which the interface conditions are not explicitly imposed. Thisformulation is based in part on the mixture formulation presented by Wangand Beckermann [68], who implement a numerical solution method for thecase of an isothermal two-phase region. The convergence of such numeri-cal schemes for nonisothermal phase change in porous media has not beenwell demonstrated in the literature, owing to the lack of exact solutions ofsuch problems. We describe the finite volume solution of the nonisothermalmixture problem, and demonstrate the validity of this method by establish-ing exact similarity solutions for reduced model problems, and presentinga numerical convergence study. We shall name our new reformulation andsolution procedure a the “M2” mixture method.

4.1 Mathematical formulation of the model

problem

Here, we present a model based on the time-dependent extensions to thesteady-state conservation equations (3.2)-(3.4), and (3.16)-(3.17). Firstly,we consider the two-phase zone 0 < z < L(t), where the interface location isnow a function of time. In this region, we have conservation of liquid mass.The mass (per unit volume) of liquid in a control volume is φρls, where φ isthe porosity of the medium, ρl is the liquid density, and s is the saturation,

Chapter 4. The M2 mixture method for the transient Udell problem 81

as before. Then, conservation of liquid mass is given by

(φρls)t + (ρlul)z = Γ, (4.1)

where, as before, u represents the Darcy velocity, and Γ is the condensationrate. Similarly, conservation of mass for the vapour phase is given by

(φρv(1 − s))t + (ρvuv)z = −Γ. (4.2)

The energy equation is now

(ρc) Tt = KTzz + hvapΓ, (4.3)

with a mass averaged product of density and heat capacity appearing. Weassume here that the dominant density-heat capacity product is that of theporous medium, such that we can neglect variations in this quantity withsaturation. Certainly, this will be true for small values of saturation near theinterface.

As in the steady-state problem, we can eliminate the condensation ratebetween the three conservation equations to give conservation of mass as

φ (ρls+ ρv(1 − s))t + (ρlul + ρvuv)z = 0, (4.4)

and(ρc) Tt = KTzz − hvap ( (φρv(1 − s))t + (ρvuv)z ) . (4.5)

Again, we have a coupled system for the two unknowns s and T in the two-phase zone 0 < z < L(t).

In the vapour-only region L(t) < z < D, there is no condensation, andconservation of mass gives

(φρv)t + (ρvuv)z = 0, (4.6)

and conservation of energy is given by the heat equation with no source term:

(ρc) Tt = KTzz. (4.7)

We now have parabolic systems in two unknowns on either side of the movinginterface z = L(t), and, as such, we require five conditions to be specifiedon the interface. The first three conditions are that saturation is zero, the


temperature is continuous, and the vapour pressure is continuous, giving theconditions

s = 0, (4.8)

[T ]+− = 0, (4.9)

[p]+− = 0, (4.10)

which are exactly the same as conditions (3.23)-(3.25). The two remainingconditions again come from conservation of mass and energy across the in-terface. These conditions require careful consideration of the effect of thenonzero velocity of the interface, and we derive these conditions in the fol-lowing subsection.

4.1.1 Modified Stefan conditions at theinterface z = L(t)

The well known Stefan condition describes conservation of energy across afreezing/melting interface moving through a body of water, say, separatingregions of liquid and solid. Following Crank [20], we have presented a deriva-tion of this condition in Chapter 2. The major assumptions which are madeare that the water density is the same in either phase, and that the water ineach phase is stationary. Clearly, the Stefan condition must be modified forproblems in which there is an interface between liquid and vapour phases,where the fluid on either side of the interface may be moving, and in caseswhere additional heat sources or sinks exist. With this in mind, we now pro-ceed to find conditions for mass and energy conservation across the interface,in terms of the interface velocity L(t).

Firstly, we consider conservation of mass across a general moving interfacewhich separates two fluids, possibly different phases, which may have differentdensities and velocities. The argument is presented for a one-dimensionalproblem, which is shown in Figure 4.1. The fluid to the left of the interfacehas density ρI, and is moving with velocity uI, while the fluid to the right ofthe interface has density ρII, and is moving with velocity uII. Suppose thatthe interface moves a distance δz in short time δt, and that it moves withvelocity L(t). Mass conservation requires that the difference between themass in the control volume [L(t), L(t) + δz] over the time interval [t, t + δt]is due to the net mass flux into the control volume during that interval. For


z

δz, δt

fluid density ρI

velocity uI

fluid density ρII

velocity uII

z = L(t)

L(t)

Figure 4.1: Mass conservation at the moving interface.

the problem shown in Figure 4.1, this gives

(ρI − ρII)δz = (ρIuI − ρIIuII)δt.

Taking the limit as δt → 0, we arrive at a condition on the velocity of theinterface, that is

(ρI − ρII)L(t) = ρIuI − ρIIuII. (4.11)

Further, in the case of flow through a porous medium which has porosity φ,the interface velocity will be given by

φ(ρI − ρII)L(t) = ρIuI − ρIIuII, (4.12)

where uI and uII are now understood to be the Darcy velocities. Now, for theUdell phase change problem, suppose we take region I to be the two-phaseregion, and region II to be the vapour-only region. The condition (4.12)holds, but we must consider mixture quantities in region I. That is, weconsider quantities associated with the mixture of liquid and vapour in thetwo-phase region. Let us define

ρI = ρls+ ρv(1 − s), the mixture density,ρIuI = ρlul + ρvuv, the mixture mass flux.

(4.13)

Then equation (4.12) gives

φ ( (ρls+ ρv(1 − s))I − (ρv)II ) L(t) = (ρlul + ρvuv)I − (ρvuv)II.


Now, in view of the continuity of temperature (4.9) and of vapour pres-sure (4.10), we have that (ρv)I − (ρv)II = 0, and hence

φ ((ρl − ρv)s)I L(t) = (ρlul + ρvuv)I − (ρvuv)II. (4.14)

To explicitly find the interface velocity, we must be careful. Specifically, thesaturation condition at the interface (4.8) is s = 0 at z = L−. The velocityL(t) must be considered as a limit of an indeterminate form. In particular,the velocity is explicitly given by

L(t) = limz→L(t)−

(ρlul + ρvuv)I − (ρvuv)II

φ(ρl − ρv)Is. (4.15)

We note that such an indeterminate form for the velocity of a moving in-terface, resulting from a mass conservation argument, also arises when con-sidering the porous medium equation, as we have seen in Chapter 2. Thecommon feature here is the degeneracy as s→ 0.

Now, for the Udell phase-change problem, we also require an energy bal-ance across the moving interface, which we will also see to be a limit ofan indeterminate form. The argument we present here is an extension ofCrank’s [20] derivation of the Stefan condition in one dimension. Our prob-lem includes an extra term due to motion and phase change in the two-phaseregion, and we refer to the energy balance as a Modified Stefan Condition.Consider the problem shown in Figure 4.2. The interface between the two-phase region and the vapour-only region in the Udell problem moves a smalldistance δz in the small time interval δt. The heat which flows into thecontrol volume during the time interval is

(K∂T

∂z

)

II

δt,

while the heat which flows out of the control volume during the time intervalis (

K∂T

∂z

)

I

δt.

Now, the heat required to evaporate the liquid which moves up to the inter-face is

(hvapρlul)δt.

These three quantities are exactly the same as for the steady-state problem.We now consider the extra heat released or required when the interface moves.


z

δz, δt

region I(two-phase)

region II(vapour)

z = L(t)

L(t)

T

Figure 4.2: Temperature profile and the moving interface.

First consider the case δz, L > 0. The additional mass which appearsin the two-phase zone after the interface has moved is φsρlδz. This mustbe exactly the amount of mass from region II which has condensed, and theheat released upon condensation is then

hvapφsρlδz.

Given these four terms, we see that the energy balance across the interfaceis given by [

K∂T

∂z

]II

I

δt+ hvapφsρlδz = hvapρlulδt.

Now, we take the limit as δz, δt→ 0 to give

hvapφsρlL(t) = hvapρlul −[K∂T

∂z

]II

I

. (4.16)

To find L(t), we note that s→ 0+ as z → L(t)−, and evaluate the limit

L(t) = limz→L(t)−

hvapρlul −[K ∂T

∂z

]III

hvapφsρl

. (4.17)


Now, if we repeat this argument for the case δz, L < 0, we get exactlythe same condition. Notice that, if we take (4.17) as the indeterminatemodified Stefan condition which determines the interface velocity, then themass balance (4.15) becomes

(ρlul+ρvuv)I−(ρvuv)II =

(ρl − ρv)I

hvap(ρlul)I −

[K ∂T

∂z

]III

hvapρl, as z → L−.

(4.18)We note here that the indeterminate modified Stefan conditions (4.15) and(4.17) can be thought of as limiting cases of the Rankine-Hugoniot condi-tions for our system. The Rankine-Hugoniot condition states that, for aconservation law of the form

At +Bz = 0, (4.19)

where B is the flux of the quantity A, then for any smooth space-timecurve z = L(t), the jumps in A and B are related to the velocity L(t) by

[A]+− L(t) = [B]+− for z = L(t), (4.20)

which expresses the conservation of A across the curve (see, for example [3]).For our problem, this may be easier to see after we have reformulated thesystem as a mixture problem, so the careful derivations of (4.15) and (4.17)are valuable here.

4.1.2 Model summary

In summary, we have a two-phase region 0 < z < L(t), in which

φ (ρls+ ρv(1 − s))t + (ρlul + ρvuv)z = 0, (4.21)

and(ρc) Tt = KTzz − hvap ( (φρv(1 − s))t + (ρvuv)z ) . (4.22)

The vapour-only region L(t) < z < D has

(φρv)t + (ρvuv)z = 0, (4.23)

and(ρc) Tt = KTzz. (4.24)


At the moving interface z = L(t), the five conditions are

s = 0, (4.25)

[T ]+− = 0, (4.26)

[p]+− = 0, (4.27)

φ ((ρl − ρv)s)I L(t) = (ρlul + ρvuv)I − (ρvuv)II, (4.28)

hvapφsρlL(t) = hvapρlul −[K∂T

∂z

]II

I

. (4.29)

At the upper boundary z = D, we impose a temperature T = T1, and haveno mass flux, so that uv = 0. At the lower boundary z = 0, we impose atemperature T = T0 (< T1), and have no mass flux, so that ρlul + ρvuv =0. Finally, we give initial profiles of saturation, temperature and pressurethroughout the entire system at time t = 0.

4.2 Fixed domain, mixture formulation

Any numerical method which is based on solutions in the disjoint domains 0 <z < L(t) and L(t) < z < D will require an implementation of the interfaceconditions. A natural approach to take is that of front tracking, which re-quires that the interface velocity be imposed explicitly. Suppose we were totake the interface condition (4.28) as the condition which defines the veloc-ity L(t). Any explicit implementation of this will require evaluation of theindeterminate form given in (4.15). Thus, any front tracking scheme requiresnot only the explicit computation of interface location at each time step, butan accurate numerical evaluation of this limit. Furthermore, extension of afront tracking scheme to higher dimensions would also require considerationof the interface geometry, and solutions to problems on irregular domains.

Clearly, front tracking is not feasible for this model problem. An alterna-tive is to reformulate the problem over the fixed domain 0 < z < D, therebyavoiding the need for explicit consideration of the complex interface physics.The interface location can be recovered, a posteriori from the solution of thetransient, fixed domain problem. That is, we aim to develop an interfacecapturing method. Here, we present a reformulation based in part on themixture model described by Wang and Beckermann [68].

The reformulation is in terms of a density over the entire pack, ratherthan saturation in just the two-phase zone. The main point is that if we


consider the water anywhere in the porous pack to be a mixture of vapourand liquid, then the density of this mixture must be continuous, even as wecross the interface. Suppose we define the mixture density ρ by

ρ = ρls+ ρv(1 − s). (4.30)

Then equations (4.21)-(4.22) reduce to the system

φρt = −(ρlul + ρvuv)z

(ρc) Tt = KTzz − hvap ( (φρv(1 − s))t + (ρvuv)z )

. (4.31)

Now, the variables uv, ul, ρv are all functions of s and T , and in particular

uv =−κµv

(1 − s)3

(∂pv∂z

+ ρvg

), (4.32)

ul =−κµls3

(∂

∂z(pv − δJ(s)) + ρlg

), (4.33)

andJ(s) = 1.417(1 − s) − 2.120(1 − s)2 + 1.263(1 − s)3, (4.34)

where pv is the vapour pressure. So if ρv, pv, s can be found as functionsof ρ and T , then (4.31) is a system in the two dependent variables ρ andT . Furthermore, given that ρ is the density of the liquid-vapour mixture,the system (4.31) is valid over the entire porous pack, rather than just thetwo-phase zone. Thus, we seek solutions to (4.31), from which we can recoverthe position of the interface between the two-phase zone and the vapour-onlyzone.

Now that we are considering a system valid over the entire domain, wecan not take the vapour pressure to be equal to the saturation pressure.Rather, the vapour pressure at a point will be equal to either the saturationpressure (in which case, the point is in the two-phase zone) or the pressuregiven by the ideal gas law for the vapour-only zone. Given values of ρ andT , a comparison of these two pressures, namely

psat(T ) = aebT , p∗(ρ, T ) =ρRT

M,

determines whether or not the vapour is fully saturated, and hence the valuesof pv, ρv and s. If p∗ < psat, then the vapour is undersaturated, so must


be in the vapour-only region. In this case, the vapour pressure p = p∗.If psat < p∗, then the vapour is fully saturated, and hence p = psat. We seethat equations (4.31), together with the algebraic constraint

pv = min

(psat(T ),

ρRT

M

)(4.35)

form a differential-algebraic system for the two variables ρ, T , which is validover the entire domain. In the next section, we describe the numerical solu-tion of this model problem, with ρ, T as the primary variables.

4.3 Computational method

Equations (4.30)-(4.35) give a coupled parabolic system on the domain 0 <z < D. Now we consider the discretization of the problem, written as

ρt =∂

∂zq

(ρ, T,

∂

∂z

), (4.36)

ρcTt + w(ρ, T )t =∂

∂zQ

(ρ, T,

∂

∂z

), (4.37)

where the fluxes q, Q are given by

q

(ρ, T,

∂

∂z

)=κ

φ

(fA(ρ, T )

∂

∂zgA(ρ, T ) + fB(ρ, T )

∂

∂zgB(ρ, T )

), (4.38)

Q

(ρ, T,

∂

∂z

)= KTz +

hvapκ

µvfC(ρ, T )

∂

∂zgC(ρ, T ). (4.39)

where

fA(ρ, T ) =ρlµls3 +

ρvµv

(1 − s)3,

gA(ρ, T ) = p,

fB(ρ, T ) =ρlµlδ,

gB(ρ, T ) = ψ(s),

fC(ρ, T ) = ρv(1 − s)3,

gC(ρ, T ) = p.


and the function w is given by

w(ρ, T ) = hvapφρv(1 − s). (4.40)

The time-dependent terms in the energy equation have been grouped to-gether, leaving the time derivative of an enthalpy-type quantity. Also, inkeeping with formulations of degenerate diffusion problems for numericalcomputation [25], we have rewritten the degenerate liquid velocity as

ul = − κ

µl

(s3pz + δ(ψ(s))z

), (4.41)

where the function ψ is given by

ψ(s) = −∫ s

ξ3J ′(ξ)dξ = 0.2415s4 − 0.6676s5 + 0.6315s6. (4.42)

This change-of-variables idea is used in preference to regularisations of thetype κrl = s3 + η, such as those seen in the fuel cell literature [50, 51]. Thenumerical convenience offered by such regularisations is due to the fact thatthey smear out the sharp interface.

Also, we have the boundary conditions

q|z=0 = qbot(t), q|z=D = qtop(t), T |z=0 = Tbot(t), T |z=D = Ttop(t). (4.43)

For the closed sand pack, the boundary fluxes are zero, and we will take theboundary temperatures to be constant in time. Now we wish to implementa numerical scheme for the solution of the system (4.36)-(4.43). Two imple-mentation options are available for explicit time-stepping. One option is touse a quasi-enthalpy method. The density ρ may be explicitly updated fromthe mass equation (4.36). Then the enthalpy E, defined by

E(ρ, T ) = ρcT + w(ρ, T ), (4.44)

may be explicitly updated from the energy equation (4.37). The tempera-ture T must then be recovered from the enthalpy by way of a nonlinear solver.A second option is to form a mass matrix, and write the system (4.31) as

(a11 a12

a21 a22

)(ρT

)

t

=

(−(ρlul + ρvuv)z

KTzz − hvap(ρvuv)z

), (4.45)


X

ρ2, τ

2

R1, T

1

Q1, q

1

X

R2, T

2

Q2, q

2

X X

ρj−1

, τj−1

X

ρj, τ

j

Rj−1

, Tj−1

Qj−1

, qj−1

X

ρj+1

, τj+1

Rj, T

j

Qj, q

j

X X

ρN+1

, τN+1

X

ρN+2

, τN+2

RN+1

, TN+1

QN+1

, qN+1

X

ρ1, τ

1

ghost cell ghost cell

Figure 4.3: Grid and staggered grid.

wherea11 = φ, a12 = 0,

together with

a21 = hvapφ

((1 − s)

∂ρv∂ρ

− ρv∂s

∂ρ

),

and

a22 = ρc + hvapφ

((1 − s)

∂ρv∂T

− ρv∂s

∂T

).

Either of these two explicit methods requires the computation of derivativesof all quantities with respect to the primary variables ρ, T , as would animplicit method. Given the inherent stiffness in the problem, here we shalluse a fully implicit scheme. Now we discretize the parabolic problem (4.36)-(4.43) by finite volumes, in order to conserve the total mass, given by

∫ D

0

ρ(z, t) dz,

in a discrete sense.Given that mass flux is given at the boundaries (4.43), we develop a

scheme which computes a discrete approximation to the solution of (4.36)-(4.37) with mass fluxes on a grid which coincides with the boundary points,and density ρ on the corresponding staggered grid. That is, we use a finitevolume scheme for updating ρj , which is the cell average density for cell j.Also, we use cell averaged temperatures. Consider Figure 4.3. Let τj and ρjbe the average values of temperature, T , and density, ρ respectively, over gridcell j. We introduce vectors T and R to represent the interpolated values oftemperature T and density ρ falling on the grid. So we have

Tj = τj+1/2 =τj+τj+1

2for j = 1, .., N + 1,

Rj = ρj+1/2 =ρj+ρj+1

2for j = 1, .., N + 1.

(4.46)


Then a fully implicit (Backward Euler), conservative, finite volume schemefor the mass equation (4.36) is given by the discretization

Fj =ρn+1j − ρnj

k− 1

h(qj − qj−1)

n+1 = 0, j = 2, .., N + 1, (4.47)

where the discrete fluxes are given by

qj = κφ

(fA(Rj , Tj)

(gA(ρj+1,τj+1)−gA(ρj ,τj)

h

)

+fB(Rj , Tj)(gB(ρj+1,τj+1)−gB(ρj ,τj)

h

) ),

j = 1, .., N + 1,

(4.48)Next, we discretize the energy equation (4.37) in a similar manner. We have

Gj = ρcτn+1

j −τnj

k+

w(ρn+1

j ,τn+1

j )−w(ρnj ,τ

nj )

k

− 1h

(Qj −Qj−1)n+1 = 0, j = 2, .., N + 1,

(4.49)

where

Qj = Kτj+1−τj

h+ hvapκ

µvfC(Rj, Tj)

(gC(ρj+1,τj+1)−gC(ρj ,τj)

h

),

j = 1, .., N + 1.(4.50)

Here, k, h are the chosen time step and grid spacing, and superscripts denotethe time level. To close the system for the 2N + 4 unknowns, τj , ρj , j = 1 :N + 2, we need four more equations. The boundary conditions give

F1 = q1 − qbot = 0,

G1 = T1 − Tbot = 0,

FN+2 = qN+1 − qtop = 0,

GN+2 = TN+1 − Ttop = 0,

(4.51)

Note that we cannot find the ghost values ρ1, ρN+2 explicitly in terms ofinterior cell values, so we just include them in the system to be solved. Thatis, we have 2(N + 2) nonlinear equations

Fj = 0, Gj = 0, j = 1, .., N + 2,


for the unknowns ρj , τj, j = 1, ..N + 2. We use a Newton method withanalytical Jacobian. In order to avoid “if” statements in the code, we replacethe nonsmooth map (4.35) with the smoothed minimum

p = minε(psat, p∗) = Hε(p

∗ − psat)psat + (1 − Hε(p∗ − psat))p

∗, (4.52)

where the smoothed Heaviside function is given by

Hε(X) =1

2

1 + tanh

(X

ε

), (4.53)

for a smoothing radius ε, and p∗ = RMρT . In practice, the smoothing radius ε

can be taken arbitrarily small, and thus, the singularity and degeneracy inthe saturation equation are retained.

4.4 Numerical results and discussion

In Figure 4.4, we show a typical result, starting from an initial uniform den-sity, with no mass flux across the boundaries. The initial temperature isuniform, and the system is subject to sudden heating up to a fixed tem-perature at the upper boundary. A number of density, temperature andsaturation profiles are shown, for increasing time, and we note that the pro-files shown approach the correct steady-state profiles. The transition to aninterface s = 0 is captured, and the interface between the two-phase andvapour regions is clearly moving to the left. In Figure 4.5, the interface po-sition, L(t) is plotted. The stepwise behaviour is due to the discretization,and the fact that we take L(t) as the first grid point to the right of s = 0.Also, we plot L(t) for three grid sizes: N = 20, 40, 80. Convergence to abase numerical solution is clear as the grid is refined. The question remains,however, of whether the numerical method indeed computes the interfacevelocity accurately. This will be addressed in the next section. Figure 4.6shows the temperature history at a point. An initial increase in temperatureis numerically the result of the sudden-heating boundary condition. Afterthis, the temperature history has a wavy, stepwise nature, in common withthe enthalpy-method solution of the Stefan problem [3].

For the results shown in Figures 4.4-4.6, small time steps are initiallyrequired, in order for the Newton iteration to converge to within the specifiedtolerance. In fact, initially, we use Forward Euler time-stepping to get past


0 0.05 0.1 0.15 0.2 0.25

20

40

60

80

Evolution to correct steady−state profiles, starting with uniform density and temperaturem

ixtu

re d

ensi

ty, ρ

0 0.05 0.1 0.15 0.2 0.25360

362

364

366

tem

pera

ture

, T

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

satu

ratio

n, s

Figure 4.4: Evolution to correct steady-state.

the initial transients, and then employ the fully implicit scheme. However,the time step required for convergence to within a fixed tolerance decreases asthe saturation profile steepens before the interface moves by a grid point. InFigure 4.7, we demonstrate the stiffness in the problem. An experiment wasperformed using implicit time stepping with adaptive time-step. Wheneverthe nonlinear solver is unable to converge to within the required tolerance,the time-step is halved. The figure clearly shows the relationship betweenthe time-step and the interface advance.


0.5 1 1.5 2 2.5 3 3.5

x 104

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

Moving interface location, with T0=360 T1=367.9727 W=13.8153 L=0.175

time t (seconds)

L(t)

N=20N=40N=80

Figure 4.5: Interface location L(t), with grid refinement.

4.5 Similarity solutions and numerical

convergence study

Numerical experiments using our mixture-based capturing method for thetime-dependent problem show that initial distributions evolve to the correctsteady-state solutions, as given by the disjoint-domain method of Chapter 3.In this section, we carefully examine the dynamics of the system to ensurethat the method indeed captures the moving interface accurately in time.Typically, the convergence of numerical methods for moving boundary prob-lems is demonstrated using initial and boundary data consistent with knownsimilarity solutions. Similarity solutions are available for the two-phase Ste-fan problem and the porous medium equation. In the literature, analyticalconvergence studies of numerical schemes have appeared for reduced, scalarmodels of two-phase flow in porous media [2, 49]. Also, analytical solutionshave been found for reduced and scalar models of phase change in porousmedia [9, 13, 34, 41, 42, 59, 60, 74]. Wang and Cheng [69] summarize analyt-ical solutions to model problems presented by a number of authors, for bothsteady-state and transient problems. These have largely been constructed byneglecting temperature and capillary effects. Wang and Cheng [69] propose


0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

360

360.5

361

361.5

362

362.5

363

363.5

364

364.5Temperature history at z=0.218

time t (seconds)

tem

pera

ture

T (

K)

Figure 4.6: Temperature history at a point.

a similarity solution for a full model including temperature and capillaryeffects, and underline the need for further analytical solutions to such prob-lems. Here, we reduce our model problem only by simplifying the coefficients,but we retain the full, nonlinear, vector problem, and construct a travellingwave solution.

4.5.1 A reduced model problem

Let us consider a model problem with reduced Darcy velocities, and themajority of coefficients equal to one. Specifically, we consider

liquid flux = −ρls3 (pz + sz) , and vapour flux = −ρvpz,

where we have simplified the Leverett function and the relative permeability,and have taken absolute permeability and viscosity to be unity. We note thatremoving the relative permeability of vapour, namely (1− s)3, will not affectthe singularity structure at s = 0, but it will allow for solutions with satu-ration s > 1. In this sense, the solutions cease to be physically meaningful.


0 1 2 3 4 5 6 7 8 9 10

x 104

10−2

10−1

100

101

102

times

tep

kM2 interface evolution and timestep

0 1 2 3 4 5 6 7 8 9 10

x 104

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0.26

time t

inte

rfac

e lo

catio

n L(

t)

Figure 4.7: Timestep and interface location.

Conservation of liquid mass reads

(ρls)t −(ρls

3(sz + pz))z

= Γ. (4.54)

Conservation of vapour mass reads

(ρv)t − (ρvpz)z = −Γ, (4.55)

where the weighting (1 − s) has been taken as 1. Again, this will not affectthe structure near s = 0. Conservation of energy is

Tt = Tzz + Γ. (4.56)

In the vapour region, we have

Γ ≡ 0, p = ρvT (ideal gas), p < psat(T ) (undersaturated), s ≡ 0,


leaving a problem for ρv, T (or p, T ). In the two-phase region, we have

p = ρvT (ideal gas), p = psat(T ) (fully saturated),

leaving a problem for s, T and Γ. At the interface z = L(t), we have fiveconditions, corresponding to (4.25)-(4.29):

s = 0, (4.57)

[T ]+− = 0, (4.58)

[p]+− = 0, (4.59)

(Mass) ρls−L(t) = −

(ρls

3(sz + pz) + ρvpz)−

+ (ρ(ρT )z)+ , (4.60)

(Energy) ρls−L(t) =

(−ρls3(sz + pz)

)− − [Tz]+− . (4.61)

Note that, in view of the ideal gas law p = ρvT , conditions (4.58)-(4.59)further imply continuity of vapour density

[ρv]+− = 0. (4.62)

Also, for convenience later, we note that (4.60)-(4.61) give

0 = − (ρvpz)− + (ρ(ρT )z)

+ + [Tz]+− . (4.63)

Then, the five conditions (4.57),(4.58),(4.62),(4.61), (4.63) can be used tosolve the problem.

For the fixed domain problem on 0 < z < D, first we define the mixturedensity

ρ = ρls + ρv, (4.64)

and then use the “M2-map”

p = min(ρT, psat(T )),

ρv = pT,

s = ρ−ρv

ρl,

(4.65)


leaving the coupled system

ρt − (ρls3(sz + pz) + ρvpz)z = 0,

(T + ρv)t = Tzz + (ρvpz)z .

(4.66)

Now, we consider forms of the saturation pressure function, psat(T ) whichyield semi-analytical travelling wave solutions.

4.5.2 Travelling wave solution for case psat(T ) = αT

Suppose the saturation pressure is a linear function of temperature, such thatpsat(T ) = αT . Then ρv ≡ α in the two-phase region. Also suppose that wehave an infinite porous medium −∞ < z < ∞, with far-field temperaturesnot yet specified. At time t, let there be an interface at z = ct separating atwo-phase region z < ct, and a vapour-only region z > ct. So the interfaceinitially is at z = 0, that is, L(0) = 0, and we seek travelling wave solutionswith speed c. Then, in the two-phase region, mass and energy conservationgive

st =(s3(sz + αTz)

)z+α2

ρlTzz (4.67)

Tt = (1 + α2)Tzz. (4.68)

In the vapour region, mass and energy conservation give

ρt = (ρ(ρT )z)z (4.69)

Tt = Tzz. (4.70)

Finally, at the interface z = L(t), the 5 conditions (4.57), (4.58), (4.62),


(4.61), (4.63) give

s = 0 (4.71)

[T ]+− = 0 (4.72)

[ρv]+− = 0, (so ρ+

v = α), (4.73)

ρls−c =

(−ρls3(sz + αTz)

)− − [Tz]+− , (4.74)

0 = −(α2Tz

)−+ (ρ(ρT )z)

+ + [Tz]+− . (4.75)

We seek travelling wave solutions to the system (4.67)-(4.70), which satisfythe interface conditions (4.71)-(4.75). In the two-phase zone, seek solutions

s(z, t) = G(ξ), T (z, t) = F1(ξ), (4.76)

and, in the vapour zone, seek solutions

ρ(z, t) = R(ξ), T (z, t) = F2(ξ), (4.77)

whereξ = z − ct. (4.78)

With c > 0 we have wavefronts moving to the right with speed c. Solutionsof the two linear heat equations are

F1(ξ) = A1 +B1e− c

1+α2 ξ, (4.79)

andF2(ξ) = A2 +B2e

−cξ. (4.80)

The constants A1, B1, A2, B2 must be consistent with the interface conditions.Continuity of temperature (4.72) gives

A1 +B1 = A2 +B2. (4.81)

Now, the saturation equation (4.67) gives

−cG′ =(G3(G′ + αF ′

1))′

+α2

ρlF ′′

1 ,


and hence

−cG = G3(G′ + αF ′1) +

α2

ρlF ′

1 + A3, (4.82)

or

G3G′ = −(cG+ α

(G3 +

α

ρl

)F ′

1 + A3

). (4.83)

The constant A3 must be consistent with the interface conditions. Condi-tion (4.74) gives

s3s−z = −(cs− + αs3T−

z +1

ρl(T+

z − T−z )

),

or

G3G′ = −(cG+ αG3(

−c1 + α2

)B1 −1

ρl(

−c1 + α2

)B1 −1

ρlcB2

).

Comparing with (4.83), we find

A3 =c

ρl(B1 − B2) . (4.84)

Then the travelling wave saturation profile G(ξ) may be found by solvingthe ODE (4.83), subject to G(0) = 0, for ξ < 0. The constant A3 is givenby (4.84), and (4.79) gives

F ′1(ξ) = − c

1 + α2B1e

− c1+α2

ξ.

Notice that we can find an analytical solution to (4.83) if B1 = 0, whichamounts to an isothermal two-phase region, with no variation in vapourpressure, and hence no vapour flow. In this special case, the exact solutionG(ξ) is given implicitly by

1

3G3−1

2B3G

2+B23G−B3

3 log

∣∣∣∣G+B3

B3

∣∣∣∣+cξ = 0, where B3 = −B2

ρl, (4.85)

andG(ξ) may be accurately computed using a root finder. In general, though,an analytical solution to (4.83) will not be available, and we compute G(ξ)using an ODE solver. However, equation (4.83) is singular at G = 0. Lettingw = G4, we have the regular problem

1

4w′ = −

(cw1/4 + α

(w3/4 +

α

ρl

)F ′

1 + A3

), w(0) = 0. (4.86)


It remains to find the density ρ in the vapour-only region. Equation (4.69)becomes

−cR′ = (R(RF2)′)′,

and hence−cR = R(RF2)

′ + A4. (4.87)

Now, to find the constant A4, we use condition (4.75) to give that, at ξ = 0,

R(RF2)′ = (1 + α2)F ′

1 − F ′2 = −cB1 + cB2,

and condition (4.73) to give −cR = −cα. Substituting in (4.87), we find

A4 = c(B1 − B2 − α). (4.88)

From (4.87), we can write

R′ = − 1

RF2(ξ)((c+RF ′

2)R + A4) , (4.89)

and then solve for R(ξ), in general, using an ODE solver. Also, we note that,for the special case B1 = B2 + α, then A4 = 0, and the exact solution R(ξ)is given explicitly by

R(ξ) =−cξ + A5

F2(ξ), where A5 = α(A1 +B1). (4.90)

Solution - SummaryWe have constructed the following travelling wave solution.

F1(ξ) = A1 +B1e− c

1+α2 ξ, (4.91)

F2(ξ) = A2 +B2e−cξ, (4.92)

and G(ξ) = w1/4(ξ), where w solves

1

4w′ = −

(cw1/4 + α

(w3/4 +

α

ρl

)F ′

1 + A3

), ξ ≤ 0, w(0) = 0,

(4.93)and finally, R(ξ) solves

R′ = − 1

RF2(ξ)((c+RF ′

2)R + A4) , ξ ≥ 0, R(0) = α. (4.94)


The constants A2, A3, A4 are given by

A2 = A1 +B1−B2, A3 =c

ρl(B1 − B2) , A4 = c(B1−B2−α). (4.95)

We have a three-parameter family of solutions to the model Udell problem.That is, a family of solutions parameterized by A1, B1, B2. These can becarefully chosen in order to have temperature increasing in both regions, andthe density decreasing to the right of the interface. A typical result is shownin Figure 4.8. Now we are in a position to compare numerical results withthe travelling wave solution.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

5

10

15

"den

sity

" R

(ξ)

Travelling wave profiles for α=0.7 ρl=5 A

1=1 B

1=−0.5 B

2=−0.83557

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−1

0

1

"tem

pera

ture

" F

(ξ)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

1

2

3

"sa

tura

tion"

G(ξ

)

ξ=z−ct

Figure 4.8: Travelling wave profiles for mixture density, temperature andsaturation.

Note: With B1 < 0 to ensure that F1 is increasing, we have F1 →−∞ as ξ → −∞. Clearly, the temperature in the two-phase region willbecome negative for some value of t, and remain negative as t increases.Our code takes the M2 comparison as ρv = min(ρ, α). This gets aroundthe fact that the vapour pressure becomes negative as temperature does.So, while comparison with this travelling wave solution serves the purposeof evaluating the performance of the numerical method, we note that the


travelling wave solutions to this reduced problem only represent anythingclose to the physical problem up to the time at which T and p becomenegative. The solution structure of the physical problem at the interface isretained, though.


In order to evaluate the performance of our capturing method applied tothe reduced Udell problem, we simply give our code initial conditions andboundary conditions consistent with the exact travelling wave solution givenby (4.91)-(4.95), for the chosen values of the parameters A1, B1, B2. In Fig-ure 4.9, we show computed interface location, noting close agreement with theexact travelling wave solutions, and apparent convergence with grid refine-ment. Now we will demonstrate the convergence of our schemes numerically.

0 0.005 0.01 0.015 0.02 0.0250

0.02

0.04

0.06

0.08

0.1

0.12Interface position for reduced Udell problem with travelling wave solutions

time, t

Com

pute

d in

terf

ace

posi

tion,

L(t

)

N=160N=320exact TW solution

c=4

c=2

Figure 4.9: Computed and exact interface location for the reduced travellingwave problem.


Although we have a second order scheme and an exact solution, we don’t ex-pect to be able to show second order accuracy. Firstly, we need an accuratemeasure of interface location. As mentioned in Chapter 2, methods are avail-able for the enthalpy solution of the Stefan problem [65], but are based ona known phase-change temperature, and will not be applicable here. In theresults shown, the interface location is simply the next grid point to the rightof s = 0. Also, the stepwise behaviour of not only the interface location, butof the temperature history at a point, introduces errors, which result in lessthan second order accuracy (see [3], for example). Even if novel methods areused to capture the interface position accurately, the stepwise temperaturehistories associated with the enthalpy method can give rise to large point-wise errors in temperature. Indeed, the L∞ norm of the temperature erroris a poor measure of accuracy, and though the L1 norm is a better measure,convergence in this norm is at first order. Bearing in mind these results,we expect similar, or worse, behaviour for our capturing method applied tothe reduced Udell problem, and no better than first order accuracy in thesolutions.

In the Tables 4.1-4.3, we show the results of a numerical convergencestudy for the finite volume, capturing method, applied to the reduced Udellproblem.

Table 4.1: Errors for reduced Udell problem, implicit (BE) time-steppingwith µ = 0.2, c = 4.

N num ‖ET‖1 factor ‖Eρ‖1 factor ‖EL‖1 factor

20 50 5.2534 E-3 9.6053 E-2 1.4120 E-240 200 1.8412 E-3 2.85 4.1025 E-2 2.34 7.3350 E-3 1.9380 800 6.7618 E-4 2.72 1.7308 E-2 2.37 3.8450 E-3 1.91160 3200 2.6595 E-4 2.54 7.2590 E-3 2.38 2.0283 E-3 1.90320 12800 1.1249 E-4 2.36 3.0324 E-3 2.39 1.0625 E-3 1.90

We have exact solutions given by the travelling waves, with speeds c =1, 2 and 4. In each case, we use implicit time-stepping, with µ = k/h2 fixed.The number of grid points is N , and num is the number of time steps. Wequantify the errors by calculating numerically the norms of the errors madein the temperature T , the mixture density ρ, and the interface location L.








Specifically, we compute the time-averaged quantities defined by

‖ET‖1 =1

num

1

N

num∑

n=1

N∑

j=1

|T (zj, tn) − Texact(zj , tn)| ,

‖Eρ‖1 =1

num

1

N

num∑

n=1

N∑

j=1

|ρ(zj , tn) − ρexact(zj , tn)| ,

‖EL‖1 =1

num

num∑

n=1

|L(tn) − ctn| .

In all three cases, we see convergence at first order. We note here that, whencomputing solutions to the two-phase Stefan problem with travelling wave


solutions, using the enthalpy method described in Chapter 2, we found thetemperature error to decrease by a factor of around 1.9-2.5 as the grid spacingis halved. Also, applying the discretization used by Evje and Karlsen [25],applied to the one-dimensional porous medium equation ut = (u3ux)x, wefound the error in u to decrease by a factor of 2.4-2.5 as the grid spacingis halved. Our capturing method for this coupled, vector, nonlinear prob-lem clearly gives convergence rates comparable with those for these scalarprototype problems.

4.5.4 Other choices for psat(T )

We briefly mention two more forms for the saturation pressure which yieldsemi-analytical travelling wave solutions which may be of interest.Case psat(T ) = αT + βThe temperature in the two-phase region, F1(ξ) may be found analytically,but it is given implicitly.Case psat(T ) = αT + βT 2

In this case, we have ρv(T ) = α + βT , which is increasing if α, β > 0. Thetravelling wave solutions in the vapour region remain the same. In the two-phase region, an analytical solution is again available for the temperatureprofile. For T (z, t) = F1(z − ct), we find that F1(ξ) satisfies

−c(1 + β)F1 = (1 + (α+ βF1)(α+ 2βF1))F′1 + A1, (4.96)

with A1 yet to be specified. Thus F1(ξ) is given implicitly by

1

2(F1 + d)2 + (a− 2d)(F1 + d) + (d2 − ad+ b) log |F1 + d|+ c(1 + β)

2β2ξ = B1,

(4.97)for some B1, where the constants a, b, d are given by

a =3α

2β, b =

1 + α2

2β2, d =

A1

c(1 + β). (4.98)

4.6 Computations in higher dimensions

One of the distinct advantages of a fixed-domain, front-capturing methodover front-tracking is that extension to higher dimensions is much more


straightforward. In this section, we return to the full mathematical modelfor the physical problem, and present some computational results in twodimensions.

4.6.1 Mathematical model

If we generalize the governing equations (4.1)-(4.3) to higher dimensions, wehave

(φρls)t + ∇.(ρlul) = Γ, (4.99)

where, as before, u represents the Darcy velocity, and Γ is the condensationrate. Similarly, conservation of mass for the vapour phase is given by

(φρv(1 − s))t + ∇.(ρvuv) = −Γ. (4.100)

The energy equation is now

(ρc) Tt = ∇.(K∇T ) + hvapΓ, (4.101)

where the Darcy velocities are functions of the saturation and the vapourpressure, and are given by

uv = −κ κrvµv

∇p, (4.102)

andul = −κ κrl

µl(∇p− δJ ′(s)∇s) , (4.103)

where µ is viscosity, κ is the permeability of the medium, κrl,rv are the relativepermeabilities, and J(s) is the Leverett function.

For our sample 2D computations, on Ω = (0, D)×(0, D), we take periodicconditions in the x direction. At the bottom boundary y = 0, take a giventemperature, and no mass flux, so that

T = T0(x, t), (ρlul + ρvuv).n = 0. (4.104)

At the top boundary y = 1, we take a higher given temperature, and no massflux, so that

T = T1(x, t) (> T0(x, t)), (ρlul + ρvuv).n = 0. (4.105)

Now, given that there is no mass flux across the boundary, our initial condi-tion fixes the total water mass W .

Now, in order to compute solutions to this problem, we simply extendthe finite volume scheme described in Sections 4.2-4.3 to the two dimensionalproblem.



0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5sa

tura

tion,

s

T1=513.3, T0=360, q=1500, s0=0.5, L=0.15

0 0.05 0.1 0.15 0.2 0.25360

380

400

420

440

460

480

500

z

tem

pera

ture

, T

Figure 4.10: Steady-state, one-dimensional profiles.

The first results we show use the steady-state results for the one-dimensionalproblem shown in Figure 4.10. The total water mass (per unit area) in thesystem is W = 35.7, and the interface is L = 0.15.

Now, we consider the two-dimensional, transient problem with uniformboundary temperatures at y = 0, D corresponding to those at z = 0, D forthe one-dimensional problem. Now we give an initial condition with a fixedwater mass (per unit length) of WD. Imposing no mass flux across the lowerand upper boundaries, and giving periodic conditions in x, we expect thesolutions to evolve towards the one-dimensional, steady-state results shown.Giving a one-dimensional initial condition gives the required result, showingagreement with the one-dimensional transient solutions at all times.

In Figures 4.11 and 4.12, we show saturation, temperature and mix-ture density plots at increasing times, for a two-dimensional initial conditionwhich has liquid water at a uniform saturation concentrated near the lowerboundary in a block as shown, and vapour in the block above it. The tem-perature is taken initially uniform, so that T (x, y, 0) = Tbot = 360, and thenthe system is subject to sudden heating at y = D. A fully implicit scheme isused, with a 20 × 20 grid.


0 0.1 0.20

0.05

0.1

0.15

0.2

0.25t=16

y

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25t=16

x

y

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25t=215

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25t=215

x

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25t=1115

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25t=1115

x

Figure 4.11: Saturation and temperature evolution. Saturation contours arein the top row, temperature contours are in the bottom row.

We note the evolution of the system towards the correct steady-state. InFigure 4.12, we can see that after a time of about 20 minutes, the interfaceis approximately at y = 0.15. The influence of the initial condition is stillapparent, but the saturation and temperature plots have roughly the familiarshape of the steady-state one-dimensional profiles.

In Figure 4.13, we plot the liquid and vapour fluxes at early and longtimes, on a saturation contour plot. At t = 16, the liquid fluxes are smalleverywhere except near the interface. At the later time, there is a largerliquid flux at the lower boundary, where the vapour condenses, and then isdriven upwards by the capillary pressure gradient. The vapour flux in thevapour region is initially large, but decreases in time as a steady-state is ap-proached. The one-dimensional steady-state solution has stationary vapourin the vapour region. As a steady-state is approached, the vapour flux be-comes essentially one-dimensional.

In Figure 4.14, we plot contours of saturation at increasing time, foran initial condition with a “blob” of two-phase fluid at uniform saturationin the centre of the domain. Again, we have sudden heating at the top,closed upper and lower boundaries, and periodic conditions in x. We see


0

0.1

0.20 0.1 0.2

0

0.2

0.4

0.6

xy

t=0

satu

ratio

n s

0

0.1

0.20 0.1 0.2

0

0.2

0.4

0.6

xy

t=1115

satu

ratio

n s

0

0.1

0.20 0.1 0.2

400

450

500

yx

tem

pera

ture

T

0

0.1

0.20 0.1 0.2

400

450

500

x

y

tem

pera

ture

T

Figure 4.12: Saturation and temperature profiles - initial condition and longtime.

the two-phase region spreading, and migrating towards the lower boundary.There is initially a large downward vapour flux in the vapour region (outsidethe blob), and condensation occurs at the cold boundary y = 0. A secondtwo-phase region thus appears near the lower boundary, creating a secondinterface which is clearly visible at t = 189. Liquid accumulates in the lowertwo-phase region, and the blob migrates downwards through the surroundingvapour region, until the two separate two-phase regions coalesce, leaving justone interface once more. Our capturing method encounters no problems uponthese topological changes.

By t = 6709, it appears that the solution is almost one-dimensional.The total mass (per unit length) in the system in Figure 4.14 is 3.7425, andthe uniform boundary temperatures are T0 = 360 and T1 = 512.7. Thus,the steady-state to which the system evolves should correspond to the one-dimensional steady-state with mass (per unit area) W = 3.7425/D = 14.734.Using our disjoint-domain method from Chapter 3, this gives s0 = 0.2, q =1000, and the interface position L = 0.104. In Figure 4.15, we plot satura-tion and temperature profiles for a fixed x in the two-dimensional problem,at a large time. Clearly, the system has almost reached the correct one-


0 0.1 0.20

0.05

0.1

0.15

0.2

0.25

0.01

0.050.08

x

ySaturation contours and liquid water flux at time t=16

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25

0.01

0.05

0.08

x

y

Saturation contours and vapour water flux at time t=16

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25

0.01

0.05

0.08

x

y

Saturation contours and liquid water flux at time t=1115

0 0.1 0.20

0.05

0.1

0.15

0.2

0.25

0.010.05

0.08

x

y

Saturation contours and vapour water flux at time t=1115

Figure 4.13: Saturation contours with liquid and vapour fluxes.

dimensional steady-state.In our final numerical experiment, we again give periodic conditions in x,

no mass flux across the lower and upper boundaries, an initially uniformtemperature distribution, at T0 = 360, and sudden heating at the upperboundary with a uniform temperature T1 = 627.8. This time, we have aninitial condition with total water mass of 7.5466, which corresponds to a one-dimensional problem with W = 7.5466/D ≈ 29.7. The corresponding one-dimensional steady-state solution has s0 = 0.58, q = 2000 and L = 0.12. Theinitial condition this time has two two-phase regions at uniform saturation,one in the shape of a rectangle, and one in a circle.

In Figure 4.16, we plot contours of saturation at increasing time. Ourmethod again captures the topological changes with no difficulty. The twotwo-phase regions spread, while a third two-phase region develops throughcondensation at the cold lower boundary. The two initial two-phase regions


t=0 t=64 t=189 t=498

t=777 t=1177 t=1435 t=1855

t=2090 t=2370 t=3088 t=3586

0 0.1 0.20

0.1

0.2

t=4014

y

x0 0.1 0.2

0

0.1

0.2

t=4850

x

y

0 0.1 0.20

0.1

0.2

t=5612

x

y

0 0.1 0.20

0.1

0.2

t=6709

x

y

Figure 4.14: Saturation contours for an initial “blob” of two-phase fluid inthe centre of the domain.

coalesce, and also the initially circular region coalesces with the two-phaseregion near the lower boundary. Eventually, the remaining vapour regionnear the lower right hand corner of the domain disappears, leaving just onetwo-phase region, and a one-dimensional steady-state is approached. In Fig-ure 4.17, we plot the saturation and temperature profiles for a fixed x inthis two-dimensional problem, at a large time. Clearly, this system has alsoalmost reached its correct one-dimensional steady-state.


0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

satu

ratio

n s

Comparison of blob solution after long time, and 1−D steady state

s(0.054,y,6709)1−D steady−state

0 0.05 0.1 0.15 0.2 0.25360

380

400

420

440

460

480

500

tem

pera

ture

T

y

T(0.054,y,6709)1−D steady−state

Figure 4.15: Long time saturation and temperature profiles at fixed x forthe spreading, migrating blob, together with one-dimensional, steady-statesolution.


0 0.1 0.20

0.1

0.2

t=0

y

0 0.1 0.20

0.1

0.2

t=9

0 0.1 0.20

0.1

0.2

t=49

0 0.1 0.20

0.1

0.2

t=99

y

0 0.1 0.20

0.1

0.2

t=160

0 0.1 0.20

0.1

0.2

t=301

0 0.1 0.20

0.1

0.2

t=444

x

y

0 0.1 0.20

0.1

0.2

t=1232

x0 0.1 0.2

0

0.1

0.2

t=3426

x

Figure 4.16: Saturation contours for an initial condition with multiple two-phase regions in the centre of the domain.


0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

satu

ratio

n s

Comparison of circle and rectangle solution after long time, and 1−D steady state

s(0.12,y,3426)1−D steady−state

0 0.05 0.1 0.15 0.2 0.25

400

450

500

550

600

tem

pera

ture

T

y

T(0.12,y,3426)1−D steady−state

Figure 4.17: Long time saturation and temperature profiles at fixed x for ini-tial multiple two-phase zone problem, together with one-dimensional, steady-state solution.

117

Chapter 5

Conclusions and future work

5.1 Summary of results

In this thesis, we have described a number of free and moving interfaceproblems, related to phase change processes in porous media. Throughoutthe thesis, we have made note of mathematical points of interest, with a viewto motivating further work.

The first result of this work is an asymptotic analysis of smoothing meth-ods applied to the one-dimensional, steady-state, two-phase Stefan problem.This appears to be new work. Smoothing techniques are often implementedin capturing methods for moving interface problems, with little mention ofthe effect they have on computational results. In fact, Crank [20] mentionedthat very little analysis of smoothing for the enthalpy function used in theenthalpy method had appeared in the literature, and this still appears to bethe case.

Also in Chapter 2, we have presented numerical convergence studies forthe well established enthalpy method for the Stefan problem, as well as amore recently described method for the Porous Medium Equation [25]. Thishas been done with a view to drawing a direct comparison between the con-vergence rates for the capturing methods for these scalar problems, and thecapturing method that we develop for the vector problem of phase change inporous media in Chapter 4.

In Chapter 3, we have modified a recently developed method for solvingthe one-dimensional, steady-state free boundary problem of a three zonesystem to be applied to a two-zone system of particular interest. The modelproblem has been extended to allow for compressible vapour. Numericalresults have been generated using this iterative disjoint-domain method.

Also in Chapter 3, the method of residual velocities [22] for solving steadyfree interface problems with linear, scalar elliptic problems on either side ofa free interface, has been extended to solve the nonlinear, vector problem ofphase-change in porous media. Numerical results show agreement with the

Chapter 5. Conclusions and future work 118

results from the iterative disjoint domain method.In Chapter 4, we have developed a numerical interface capturing method

for a model problem of time-dependent two-phase flow with phase change inporous media. We have allowed for a nonisothermal two-phase region, ratherthan making the popular assumption of an isothermal region. This will allowfor computations in particular settings where thermal effects are important,such as fuel cell electrodes.

Numerical solutions found using this capturing method are seen to con-verge in time to the correct steady-state solutions generated using the methodof Chapter 3. These benchmark steady-state solutions have been particularlyuseful in validating the results of the capturing method.

We have found similarity solutions to a model vector problem for phasechange in porous media. The singularity, degeneracy and coupling in themathematical model have been retained, and we have demonstrated conver-gence of our numerical results to the exact travelling wave solutions, thusshowing that the capturing method recovers an interface moving at the cor-rect velocity. To our knowledge, this is the first such convergence study for afull, coupled model. Convergence rates are comparable with those presentedin Chapter 2 for the simpler, scalar problems.

The implementation of the capturing method has been extended to thetwo-dimensional problem, and we have demonstrated that our method caneasily deal with multiple interfaces, and topological changes such as disap-pearing interfaces. This is a distinct advantage over front-tracking methods.Also, no grid refinement near the interface is required by our method, whichis an advantage over a recently described numerical capturing method fora related problem in the fuel cell literature [10]. Convergence to the cor-rect steady-state solutions is once again demonstrated using our benchmarksolutions.

5.2 Future work

We suggest using the ideas presented in Chapter 2 as part of an asmymptoticanalysis of smoothing techniques applied to the time-dependent Stefan prob-lem. This will involve the analysis of smoothing strategeies applied to thediscontinuous enthalpy functions, and possibly an accompanying analysis ofthe relationship between the smoothing parameters and the computationalgrid spacing.


In Chapter 3, the method of residual velocities has been used for the one-dimensional porous medium problem. Future work which we have suggestedincludes an implementation of the method applied to the two-dimensional.The analysis of the residual velocity method has so far been applied to prob-lems of Laplacian type [19, 22], and we suggest using the model problemdescribed in Subsection 3.3.2 as a starting framework for the analysis of thefull Udell problem.

The capturing method that we have developed in Chapter 4 appears tobe robust, and no grid refinement has been necessary in our computations.We have, however, used an adaptive time-stepping method. Future work willinvolve looking at strategies for this adaptive time stepping.

Our computational method has been implemented for the so-called Udellproblem of phase change in porous media. The model problem which wehave considered is quite general, and future work will include applying thiscapturing method to specific physical and industrial problems of interest.In particular, we will look at model problems of phase change in fuel cellelectrodes and oil reservoirs.

Also, we will consider modified models of phase change which relax thesaturation assumption which we have used throughout this thesis. We brieflydescribe such a model problem below, and some ideas concerning its solution.

5.2.1 The “Big-H” regularisation

We take this idea from the fuel cell modelling literature (see, for example, [10,21, 50]), and refer the reader to Wang [67], who discusses the computationalconvenience offered by this regularisation. The vapour is not assumed to beat saturation in order for phase change to occur. Instead, any oversaturatedvapour is supposed to condense at a very large rate, proportional to thedegree of oversaturation. Undersaturated vapour may not condense. In thetwo-phase region, if vapour is undersaturated, then evaporation of liquidwater occurs at a large rate, proportional to the degree of undersaturation.On the other hand, if vapour is undersaturated and no liquid is present, thenno phase change occurs. We take the condensation rate to have the followingform:

Γ =

H+(pv − psat(T )) pv ≥ psat(T )H−s (pv − psat(T )) pv < psat(T )

. (5.1)

Here, H+ and H− are large constants, which account for the large conden-sation rates. The factor of s precludes the possibility of evaporation if there


is no liquid present. Now the governing equations for the model problemare the system (4.31), together with the condensation rate (5.1). Theseconstitute three equations for three unknowns s, T and p on the fixed do-main 0 < z < D, rather than the two equations in two unknowns when we usethe saturation assumption. This idea has been used in the fuel cell literature,but computations have only been performed under simplifying assumptions,such as isothermal conditions, or with computational regularisations or localgrid refinement [10]. The method we have described in this thesis does notrequire any of these simplifications. The relation between our model andthis “Big-H” model in the limit of large H+, H− is of interest to us, and inparticular, how the size of these parameters affects the computations.

We propose to implement this method for the reduced model problemdescribed in Subsection 4.5.1. An asymptotic analysis of this method willappear in future work, where we take H+ = H− = 1/ε. Preliminary analysislooking for travelling wave solutions suggests a corner layer will be introducedwhich will smear out the sharp interface, but that the sharp interface solutionwill be approached as ε→ 0. This is continuing work.

121

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